macroeconomic effects on the stock market

UCD Michael Smurfit Business School

FIN40020

Financial Econometrics

Group Project

Macroeconomic effects on the stock market

Chan, Wing Fei 14201368

Kevin Walsh 05595304

Ye, Ying 14203002

Zhou, Xuan 13209238

Zhu, Qing Ying 14203262

Statement

we declare that all material included in this project is the end result of our own work and that

due acknowledgement has been given in the bibliography and references to all sources be

they printed, electronic or personal

3rd December 2014


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Contents CHAPTER 1: INTRODUCTION ..................................................................................................... 4

1.1 Introduction ........................................................................................................................... 4

1.2 Motivation ............................................................................................................................... 5

CHAPTER 2: LITERATURE REVIEW ......................................................................................... 6

2.1 Introduction ........................................................................................................................... 6

2.2 Chen, Roll and Ross, (1986) ............................................................................................. 6

2.3 Other important research ................................................................................................. 8

2.3.1 Shanken and Weinstein (2006) ............................................................................. 8

2.3.2 Lamont, (2000)............................................................................................................. 8

2.3.3 Ferson & Harvey, (1993) .......................................................................................... 8

2.3.4 Cutler, Poterba and Summers, (1989) ................................................................. 9

2.3.5 Mcqueen & Roley, (1993), Boyd, Jagannathan & Hu, (2001) ..................... 9

2.3.6 Hamilton & Susmelb, (1994) ................................................................................... 9

2.3.7 Fama, (1981), (1990) .............................................................................................. 10

2.3.8 Schwert, (1989) ........................................................................................................ 10

CHAPTER 3: METHODOLOGY .................................................................................................. 11

3.1 Data Collection ................................................................................................................... 11

3.2 Data Processing ................................................................................................................. 12

3.3 Methodology ....................................................................................................................... 13

3.3.1 Ordinary Least Square ............................................................................................ 13

3.3.2 Model Specification .................................................................................................. 13

3.3.3 Autocorrelation ......................................................................................................... 16

3.3.4 Heteroskedasticity ................................................................................................... 17

3.3.5 Multicollinearity ....................................................................................................... 20

3.3.6 Exogeneity ................................................................................................................... 21

Chapter 4: DATA ANALYSIS ...................................................................................................... 23

4.1 Descriptive Statistic ......................................................................................................... 23

4.2 Model Construction .......................................................................................................... 25

4.3 Preliminary Examination of Regression Residuals ............................................. 27

4.4 Model Specification .......................................................................................................... 30

4.4.1 Information Criterions ........................................................................................... 30

4.4.2 Durbin Watson d Test ............................................................................................. 30

4.4.3 Ramsey RESET Test ................................................................................................. 31

4.4.4 F-Test ............................................................................................................................ 32


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4.5 Description of Empirical Model ................................................................................... 33

4.6 Diagnostic Testing ............................................................................................................ 34

4.6.1 Autocorrelation ......................................................................................................... 34

4.6.2 Heteroskedasticity ................................................................................................... 36

4.6.3 Mutlicollinearity ....................................................................................................... 40

4.6.4 Exogeneity ................................................................................................................... 43

CHAPTER 5: CONCLUSION ........................................................................................................ 44

5.1 Discussions .......................................................................................................................... 44

5.2 Summary of Statistical Analyses ................................................................................. 44

5.3 Conclusion ........................................................................................................................... 46

References ....................................................................................................................................... 48

Appendices ...................................................................................................................................... 51


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CHAPTER 1: INTRODUCTION

1.1 Introduction

The relationship between macroeconomic factors and stock market returns has been a

prominent topic of academic research over the past number of decades. Some financial

theory suggests that macroeconomic variables should systematically affect stock

market returns. Asset prices are commonly believed to react sensitively to economic

news and daily practice seems to support the view that individual asset prices are

influenced by a wide variety of unanticipated events and that some events have a more

prevalent effect on asset prices than do others. Some fundamental macroeconomic

variables such as exchange rate, interest rate, industrial production and inflation have

been argued to be the determinants of stock prices. It is believed that government

financial policies and macroeconomic events have large influence on general economic

activities including the stock market. This has motivated many researchers to

investigate the dynamic relationship between stock returns and macroeconomic

variables

This report sets out to establish how macroeconomic factors affect returns on the S&P

500 Index. We expand on previous research by Chen, Roll and Ross, 1986 by modelling

equity returns as functions of macro-economic variables and asset returns. We ran

Ordinary Least Squares (OLS) regression to test the significance of the economic

variables on the S&P 500 index. To make sure OLS provided a valid result, we had to

insure that the tests were in line with gauss Markov Theorem. We also had to perform

tests to insure that we avoided problems such as model specification, autocorrelation,

heteroskedasticity, mutlicollinearity and exogeneity. In our tests, the S&P500 is our

dependent variable and our independent variables include Monthly industrial

Production, Change in expected Inflation, Unexpected Inflation, Risk Premium, Term

Structure, oil price changes and concumption expenditure. Whilst Chen, Roll and Ross

used data from 1953 - 1978, we used data from 2007 - 2011. Our data sample is 60 and

we used a monthly timeframe. In this paper, we investigate the null hypothesis that each

of the macroeconomic factors is not related to any one of the common stock factors.


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Our research will help investors by showing them how macroeconomic factors might

predict the future returns of the S&P 500 Index. It could also provide information to

governments regarding how decisions they make concerning economic policy will

affect the stock market.

1.2 Motivation

Chen, Roll and Ross (1986) state that “A rather embarrassing gap exists between the

theoretically exclusive importance of systematic "state variables" and our complete

ignorance of their identity. The co movements of asset prices suggest the presence of

underlying exogenous influences, but we have not yet determined which economic

variables, if any, are responsible” (Chen, Roll and Ross, 1986)

Our aim was to investigate the effect of macroeconomic determinants on the

performance of the S&P 500 using monthly data over the period from 2007 – 2011 for

seven macroeconomic variables. We used Chen, Roll and Ross’s 1986 model which

determined seven economic variables that could be a source of systematic risk. The

empirical model of our report uses variables including, Monthly industrial Production,

Change in expected Inflation, Unexpected Inflation, Risk Premium, Term Structure,

Oil price and Consumption expenditure.

We believe that Chen, Roll and Ross’s 1986 model is outdated and the results conveyed

by their research could differ over time given the advances in technology and ease of

access of information now, compared with back then. Retesting their model using more

recent data will give investors in the market as well as governments more pertinent

information regarding macroeconomic factors that affect stock prices.

The rest of this paper is organised as follows; Chapter 2 of this report is a Literature

Review, we review past literature on the subject and explain how it corresponds to our

research. Chapter 3 is the Methodology. In this chapter, we then explain the techniques

used to measure unanticipated movements in the proposed variables Chapter 4 reports

the results from our tests and finally Chapter 5 is the Conclusion. This section briefly

summarises our findings and suggests some directions for future research.


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CHAPTER 2: LITERATURE REVIEW

2.1 Introduction

The relationship between macroeconomic variables and a developed stock market is

well documented in literature. There is vast amounts of research concerned with the

forces that determine the prices of risky securities, and there are a number of competing

theories of asset pricing. These include the original capital asset pricing models

(CAPM) of Sharpe (1964), Lintner (1965) and Black (1972), the intertemporal models

of Merton (1973), Long (1974), Rubinstein (1976), Breeden (1979), and Cox, J.,

Ingersoll, J., Ross, S. 1985, and the arbitrage pricing theory (APT) of Ross (1976). The

most relevant study relating to our paper is the Chen, Roll and Ross 1986 paper, which

we have based our model on. Chen, Roll and Ross were the first, in a series of studies,

to employ specific macroeconomic factors as proxies for the state variables in the

Arbitrage Pricing Model

2.2 Chen, Roll and Ross, (1986)

This paper tests whether innovations in macroeconomic variables are risks that are

rewarded in the stock market. They note that financial theory suggests that macro-

economic variables such as the spread between long and short interest rates, expected

and unexpected inflation, industrial production, and the spread between high- and low-

grade bonds, should all systematically affect stock market returns The authors believed

however, that there was little research completed regarding which macroeconomic

events are likely to influence all assets.

Tests

The set of variables they used to undertake the tests were, Industrial production,

Inflation, risk premium, the term structure, market indices, consumption and finally oil

prices.


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They used a version of the Fama,MacBeth (1973) model to determine whether the

identified economic state variables are related to the underlying factors that explain

pricing in the stock market.

This involved:

Choosing a sample of assets

Estimating the assets exposure to the economic state variables by regressing

their returns on the unanticipated changes in the economic variables over an

estimation period

The resulting estimates of exposure (betas) were used as the independent

variables in 12 cross-sectional regressions, with asset returns for the month

being the dependent variable. This determined the risk premium associated

with the state variable and the unanticipated movement in the state variable for

that month.

The first two steps were then repeated for each year in the sample. This yielded

a time series of estimates of its associated risk premium for each macro variable.

The time-series means of these estimates were then tested by a t-test for

significant difference from zero.

Conclusion

They find that these sources of risk are significantly priced and that neither the market

portfolio nor aggregate consumption are priced separately. They also find that oil price

risk is not separately rewarded in the stock market.

Several of the economic variables that they chose to use were found to be significant in

explaining expected stock returns, most notably, industrial production, changes in the

risk premium and twists in the yield curve.

They found that even though a stock market index such as the value-weighted New

York Stock Exchange index, explains a significant portion of the time-series variability

of stock returns, it has an insignificant influence on pricing when compared against the

economic state variables.


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They also examined the influence on pricing of exposure to innovations in real per

capita consumption and found that the consumption variable was never significant.

Finally,

Finally they examined the impact of an index of oil price changes on asset pricing and

found no overall effect. They conclude that stock returns are exposed to systematic

economic news, that they are priced in accordance with their exposures, and that the

news can be measured as innovations in state variables whose identification can be

accomplished through simple and intuitive financial theory.

They report that the null hypothesis that each of the macroeconomic factors is not

related to any one of the common stock factors is rejected in every case, except for the

case of inflation.

2.3 Other important research

2.3.1 Shanken and Weinstein (2006)

Shanken and Weinstein (2006) re-examined the pricing of the Chen, Roll, and Ross’s

macro variables and found them to be surprisingly sensitive to reasonable alternative

procedures for generating size portfolio returns and estimating their betas. They

concluded that Industrial Production was the only significant economic factor that

affects stock markets.

2.3.2 Lamont, (2000)

Lamont,(2000), seeks to identify priced macro factors by determining whether a

portfolio Constructed to track the future path of a macro series earns positive abnormal

returns. He concludes that portfolio’s that track the growth rates of Industrial

Production, Consumption and Labour Income, earn abnormal positive returns. While

the portfolio that tracks the Consumer Price Index does not.

2.3.3 Ferson & Harvey, (1993)

Ferson & Harvey, 1993 investigate the predictability in national equity market returns,

and its relation to global economic risks. They show how to consistently estimate the


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fraction of the predictable variation that is captured by an asset pricing model for the

expected returns. They used a model in which conditional betas of the national equity

markets depend on local information variables, while global risk premia depend on

global variables. They examined single and multiple-beta models, using monthly data

for 1970 to 1989. They found that models can capture much of the predicted variation

in a sample of returns for 18 countries.

2.3.4 Cutler, Poterba and Summers, (1989)

Cutler, Poterba & Summers, 1989 examined the extent to which ex-post movements in

aggregate stock prices could be attributed to the arrival of news. They examined the

fifty largest one-day returns on the S&P 500 index over the period from 1946 through

1987. They found that Industrial Production Growth is significantly positively

correlated with real stock returns over the period 1926 – 1986 but not in the 1946 –

1985 sub period. They also found that Inflation, money supply and long-term interest

rates did not affect stock returns.

2.3.5 Mcqueen & Roley, (1993), Boyd, Jagannathan & Hu, (2001)

There are studies that suggest that surprise announcements about macroeconomic

factors may yield different results depending on the period of the business cycle that

we are currently in. Mcqueen and Roley, 1993 suggest that an increase in employment

may be a bullish sign as the economy emerges from a recession but may be a bearish

sign near a cyclical peak. Boyd, Jagannathan and Hu, 2001 also prescribe to this theory.

They examined the impact of surprise unemployment announcements on the S&P 500

over the 1948 – 1995 period. They conclude that high surprise unemployment raises

stock prices during an economic expansion but lowers stock value during a contraction.

2.3.6 Hamilton & Susmelb, (1994)

Hamilton & Susmelb, 1994 found that extremely large shocks, such as the October

1987 crash, arise from different causes and have different consequences for subsequent

volatility than small shocks. They explore this possibility with U.S. weekly stock

returns, allowing the parameters of an ARCH process to come from one of several


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different regimes, with transitions between regimes governed by an unobserved

Markov chain. They estimate models with two to four regimes in which the latent

innovations come from Gaussian and Student f distributions. They conclude that macro

conditions significantly affect equity returns.

2.3.7 Fama, (1981), (1990)

A study of the relationships between stock prices and real activity, inflation, and money

conducted by Fama in 1981 shows a strong positive correlation between common stock

returns and real variables. Another study by Fama, 1990 argues that because equity

prices reflect expected future cash flows, equity price changes should predict future

macro conditions.

2.3.8 Schwert, (1989)

Schwert, 1989 analyses the relation of stock volatility with real and nominal

macroeconomic volatility, economic activity, financial leverage, and stock trading

activity using monthly data from 1857 to 1987. He found that financial asset volatility

helps to predict future macroeconomic volatility

And finally Adebiyi et al. (2009) found that there was a causal relationship between oil

price shocks and real exchange rates, to stock prices.

As can be seen from the above literature review, there have been many contradictory

studies regarding the topic. We hope that our tests can provide some clarity to the issue

and encourage further research regarding macroeconomic factors effect on stock

returns.


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CHAPTER 3: METHODOLOGY

In this section, we will be clarifying the data collection and the methodology used in

this paper. Our main objective is to examine the relationship between various

macroeconomic factors and S&P 500 stock index. We are using ordinary least square

to run regression analysis, and a few diagnostic tests are carried out to make sure our

linear regression model is in line with Gauss–Markov theorem. In order to compute this

regression model, software from MathWork is adapted in our paper: MATLAB.

3.1 Data Collection

Secondary data was collected for the following analysis. Historical closing price of

S&P 500 stock index and macroeconomic variables data were obtained from Thomson

Reuters Datastream. We are reexamining the same model from Chen, Roll, and Ross

(1986) and applying it to recent data. The data collected covered a 5 year period, starting

from January 2007 to December 2011 in a monthly frequency with a total of 60

observations.

Table 3.1 Definitions of Variables

Symbol

Variable

Definition (Data stream: source code)

I Inflation Log Relative of U.S. Consumer Price Index

(USCONPRCE)

TB Treasury-Bill

Rate US T-BILLS BID YLD 1M (TRUS1MT)

LGB

Long-Term

Government

Bonds

US TREASURY YIELD ADJUSTED TO

CONSTANT MATURITY - 20 YEAR

(USGBOND.)

IP Industrial

Production

US INDUSTRIAL PRODUCTION - TOTAL

INDEX (USIPTOT.G)


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Baa Low-Grade

Bonds

US CORP BONDS MOODYS SEASONED

BAA (D) - MIDDLE RATE

(FRCBBAA)

OG Oil Prices Log Relative of US PPI - CRUDE

PETROLEUM (USPCIPCOF)

S&P S&P Composite

Index

S&P 500 COMPOSITE - PRICE INDEX

(S&PCOMP)

EXP Consumption

Expenditures

US PERSONAL CONSUMPTION

EXPENDITURES (AR) CURA (USPERCONB)

3.2 Data Processing

After we collected the data we needed from the reliable data source, Thomson Reuters

Datastream. We rearranged it to transform it into the variables we needed to run the

Chen, Roll, and Ross (1986) model. The formulas for data transformation can be

observed in Table 3.2

Table 3.2 Data processing

Symbol Variable Derived Series

S&P S&P composite index -

MP Monthly Industrial

Production ln[𝐼𝑃(𝑡)/𝐼𝑃(𝑡 − 1)]

DEI Change in Expected

Inflation E[I(t + 1)|t] - E[I(t)|t - 1]

UI Unexpected Inflation I(t) - E[I(t)|t - 1]

UPR Risk Premium Baa(t) - LGB(t)

UTS Term Structure LGB(t) - TB(t - 1)

OP Oil prices 𝑙𝑛 [𝑂𝑃(𝑡)/𝑂𝑃(𝑡 − 1)]

EXP Personal Consumption

Expenditures 𝑙𝑛 [𝐸𝑋𝑃(𝑡)/𝐸𝑋𝑃(𝑡 − 1)]


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3.3 Methodology

3.3.1 Ordinary Least Square

Ordinary least square (OLS) is a method that generates the estimators, which minimize

the squared distance between the regression value and the real value, for the regression

model (Leng, Zhang, Kleinman and Zhu, 2007). Moreover, “The so-called Gauss-

Markov theorem states that under certain conditions, least-squares estimators are ‘best

linear unbiased estimators’ (BLUE), ‘best’ meaning having minimum variance in the

class of unbiased linear estimators”. Furthermore, the certain conditions are: First, the

regressand can be calculated by the linear function of regressors; second, the mean of

the disturbance term is zero; third, the variance of the disturbance term is constant;

fourth, the covariance of the disturbance term is zero; fifth, the distribution of residuals

of the regression model should be normal. However, even if the residuals are not

normally distributed, it will not affect the accuracy of regression results. Lastly, the

independent variables are non-stochastic (Chipman, 2011).

3.3.2 Model Specification

According to our analysis, under the Gauss-Markov theorem, the classical linear

regression model should have accurate variables and the form of function.

3.3.2.1 Akaike & Schwarz Information Criteria

The Akaike information criterion (AIC) is defined as:

AIC = 2K − 2 ln(L)

Where K is the number of parameters, L is the function of maximum likelihood. AIC

suggests that the best model is the one that fits data well without over fitting. Therefore,

the regression model with the smallest AIC is the best fitting model (Bozdogan, 2000).


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The Schwarz information criterion, which is also named Bayesian Information

Criterion (BIC), is defined as:

BIC = −2 ln(L) + ln(n)K

Where K is the number of parameters, L is the function of maximum likelihood and n

is the sample size. Similar to AIC, BIC is the method to find the model that can fit the

data best, the smaller the BIC, the better the model (Liddle, 2007).

3.3.2.2 Durbin Watson D Test

Durbin Watson test (DW) is a method to test the autocorrelation. Assuming that the

error term of a regression model can be described as:

Ut = ρUt−1 + ε

If ρ equals 0, then the model does not have autocorrelations. We can use the d value to

test the null hypothesis, which assumes that ρ equals 0. The d value of DW test is

defined as:

d = ∑(Ut − Ut−1)2 / ∑ Ut2 ≈ 2(1 − ρ)

The d values range from 0 to 4. If the value of d closes to 4 or 0, then the disturbance

terms of the model have negative or positive correlation respectively (Femenias, 2005),

which means that essential variables are omitted.

3.3.2.3 Ramsey Reset Test

“The reset test proposed by Ramsey is a general misspecification test, which is designed

to detect both omitted variables and inappropriate function form. The reset test is based

on the Lagrange Multiplier principle and usually performed using the critical values of

the F-distribution (Schukur and Mantalos, 2004). Specially, in a reset test, the null


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hypothesis is that the regression model is a specification model. Considering the

following function:

y = ∂x + γ1y2 + ⋯ + γk−1yk + ϵ

By using the F-test, if the value of γn is significantly different from zero, which

indicates that the non-linear regressors can affect the regressand, the model is a

misspecification model (Ramsey, 1969).

3.3.2.4 Davidson & Mackinnon Test

The Davidson and Mackinnon J test is an approach to choose the best model among

non-nested competing models by building a new nested model, which includes all non-

nested models. For example, yn is the value calculated by model A and zn is the value

calculated by model B. Then use the yn and znas variables and add them into a new

model C. Moreover, use F test to test the statistically significant of the coefficient of

ynandzn. Assuming that the coefficient of yn and zn are β and α respectively, then the

following conclusions can be draw: If β is significantly different from 0 and α is not

significantly different from 0, then we choose model A; If β is not significantly different

from 0 but α is significantly different from 0, then model B is better than model A; If

both coefficients are significantly different from 0 (or not), then the J test cannot give

a specific answer (Davidson and Mackinnon, 1981). Furthermore, the J test is not

appropriate for small sample, under which the J test may reject the null hypothesis when

it is true (Godfrey and Pesaran, 1983).

3.3.2.5 Jarque-Bera Test

Jarque-Bera test is a method to test whether the sample is normally distributed.

Specially, the skewness and kurtosis of a normal distribution are equal to 0 and 3

respectively. Therefore, the closer the skewness and kurtosis of sample distribution to

0 and 3respectively, the closer the sample distribution to the normal distribution. The

JB test is defined as:

JB =n

6(s2 +

1

4(k − 3)2)


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Where n is the sample size, K is the Kurtosis, S is the Skewness. The JB statistic is

asymptotic to a chi-squared distribution if the sample is derived from a normally

distributed population. The null hypothesis of JB test is that the sample is normally

distributed. The rejection of the null hypothesis means that the distribution of sample

is not normally distributed (Jarque and Bera, 1980).

3.3.3 Autocorrelation

Autocorrelation is the disturbance terms of a regression model correlated to each other.

Under the autocorrelation, the estimators of the model are still un-biased and consistent,

but they may not be efficient any more (Brindley, 2008).

3.3.3.1 Durbin-Watson Test

As shown in the model specification section, Durbin Watson test (DW) is a method to

test the autocorrelation and the d values of the DW test range from 0 to 4. Moreover,

there is an upper bound (du)and a lower bound (dl) in the test. If4 − dl < d < 4,

or0 < d < dl , the model has serial correlation. Ifdu < d < 4 − du , the disturbance

terms do not correlate with each other. Otherwise, the d test cannot provide an answer

(Femenias, 2005).

3.3.3.2 Breusch-Godfrey Test

The Breusch-Godfrey test is a more powerful test for examining serial correlation than

the Durbin Watson test, since it can be used on autoregressive model and to test higher

order autocorrelation. The null hypothesis of the B-G test is that there is no serial

correlation, under which the distribution of (n − p)R2is asymptotic to a chi-squared

distribution. The rejection of the null hypothesis indicates that there is serial correlation

(Breusch, 1978 and Godfrey, 1978).


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3.3.3.3 Remedies

If the autocorrelation is caused by model misspecification, then we can specify the

model and avoid the autocorrelation. If not, we can continue to use the OLS model or

use some remedial models. This paper considers two remedial methods: The Cochrane-

OrcuttAR(1) adjusted OLs regression and the heteroskedasticity and autocorrelation

consistent standard errors, define as following:

Cochrane–Orcutt Estimation

If an OLS model has autocorrelation, then the model is not appropriate to be used. In

this situation, the generalized least square (GLS) model can be used to substitute the

OLS model and avoid the autocorrelation.

Newey-West Estimator

If the autocorrelation cannot be eliminated, then the heteroskedasticity and

autocorrelation (HAC) consistent standard errors, which is introduced by Newey

(1986), can be used to solve the problem of autocorrelation and heteroskedasticity.

3.3.4 Heteroskedasticity

3.3.4.1 Preliminary Examination of the Residuals

We test the normality of residuals using the following 2 methods:

Informal Method: Histogram and Q-Q Plot

Histogram is a graphical representation of the distribution of data and is an estimate of

the probability distribution of a continuous variable.


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Q–Q plot is a graphical method for comparing two probability distributions by plotting

their quintiles against each other. If the two distributions being compared are similar,

the points in the Q–Q plot will approximately lie on the line y = x.

Formal Method: Jarque-Bera Test

The Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness

and kurtosis matching a normal distribution:

JB is defined as:

Where n is the number of observations (or degrees of freedom in general); S is the

sample skewness, and K is the sample kurtosis.

In order to test whether they’re heteroskedasticity in the residuals, we use following

three methods:

3.3.4.2 Goldfeld-Quandt Test

Goldfeld–Quandt test is a method used to check for homoscedasticity in regression

model (Thursby, 1982), and there are 4 steps:

Step 1. Order the observations, Y, according to the value of X, beginning with the

lowest of the X values.

Step 2.Omit c central observations (c = (1/5)*n). Remaining two groups of (n-c)/2

observations.

Step 3. Compute the residual sum of squares from the 1st and 2nd groups

Step 4. Compute the ratio Lambda = [RSS2/df]/ [RSS1/df], where df = [(n-c)/2]-k, k is

the number of parameters to be estimated in each regression including the intercept, n

is the sample size.

http://en.wikipedia.org/wiki/List_of_graphical_methods

http://en.wikipedia.org/wiki/Probability_distribution

http://en.wikipedia.org/wiki/Goodness-of-fit

http://en.wikipedia.org/wiki/Skewness

http://en.wikipedia.org/wiki/Kurtosis

http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/Skewness

http://en.wikipedia.org/wiki/Kurtosis

http://en.wikipedia.org/wiki/Homoscedasticity


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Under the 5% significance level, if the p-value is larger than 0.05, we accept the null

hypothesis that the residuals in the population regression function are homoscedastic.

Otherwise, we accept the alternative hypothesis that the residuals in the population

regression function are heteroskedastic.

3.3.4.3 White’s General Heteroscedasticity Test

Is easier to implement and does not rely on the normality assumption, it also follows 4

steps (Koenker and Bassett, 1982):

Step 1. Estimate the linear regression model.

Step 2. Obtain the R2 from the auxiliary regression.

Step 3. Under the null of homoskedasticity it can be shown that

n. 𝑅2~𝑥𝑑𝑓2

Step 4. If one rejects the null hypothesis then there is heteroskedasticity at the selected

confidence level.

3.3.4.4 Breusch Pagan Heteroscedasticity Test

This test follows the simple three-step procedure:

Step 1: Apply OLS in the model and compute the regression residuals.

Step 2: Perform the auxiliary regression

Step 3: The test statistic is the result of the coefficient of determination of the auxiliary

regression in Step 2 and sample size with:

The test statistic is asymptotically distributed as 𝜒2(𝑝 − 1) under the null hypothesis

of homoscedasticity (Lyon and Tsai, 1996).

http://en.wikipedia.org/wiki/Asymptotic_distribution

http://en.wikipedia.org/wiki/Null_hypothesis


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3.3.5 Multicollinearity

Because there are multiple predictors in our model, if these predictors are correlated

and give redundant information, in other words, the model input variables are not

independent of one another: 𝑥𝑖 ≈ ∑ 𝑥𝑖𝑗≠𝑖 𝛼𝑗.

Perfect model:

𝜆1𝑋1 + 𝜆2𝑋2 + ⋯ + 𝜆𝐽𝑋𝑗 = 0

Less than perfect:

𝜆1𝑋1 + 𝜆2𝑋2 + ⋯ + 𝜆𝐽𝑋𝑗 + 𝜈𝑖 = 0

Multicollinearity has a strong effect on the size of the regression coefficients, and

sometimes it can cause coefficients to have opposite signs. The reason for the event of

multicollinearity problem could lead to the wrong usage of dummy variables for an

equation or including a similar variable which is highly correlated (Gujariti & Porter,

2009). In this section, these methods are used:

1. Variance Inflation Factor (VIF).

2. Tolerance (TOL).

3. Condition Indices (CI)

3.3.5.1 Variance Inflation Factor (VIF)

As we know, the variance of an OLS estimator is:

Var(𝛽��) =𝜎2

∑(𝑥𝑖 − ��)2 (1

1 − 𝑅𝑖2)

Where𝛽��is the partial regression coefficient of 𝑥𝑖 and 𝑅𝑖2 is the R2 in the regression of

𝑥𝑖on the remaining (K-2) regressors.

The variance inflation factor, or VIF, is a measure of the multicollinearity of a given

predictor variable. For a variable i, the VIF is calculated by computing the R2 from a

regression with i and all other predictor variables. The function is:


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𝑉𝐼𝐹𝑖 =1

1 − 𝑅𝑖2

Larger values of VIF indicate more multicollinearity. In other words, the larger the

VIF, the larger the standard error of the regression coefficient for variable i.

The tolerance is another measure of the multicollinearity for a given variable i.

3.3.5.2 Tolerance (TOL)

The inverse of the VIF is called tolerance (TOL).

𝑇𝑂𝐿𝑖 =1

𝑉𝐼𝐹𝑖

The smaller the tolerance, the larger the standard error of the regression coefficient

for variable i. In other words, smaller values of tolerance indicate more

multicollinearity.

3.3.5.3 Condition Indices (CI)

And the Condition indices are computed by finding the eigenvalues of the correlation

matrix of the variables in the study. It gives an estimate of multicollinearity for each

successive eigenvalue:

𝐶𝐼𝐼 = √𝜆𝑚𝑎𝑥

𝜆𝑖

High variance decomposition proportions (>0.5) for two or more estimated regression

coefficient variances corresponding to the same small singular value associated with

each high condition index will identify the covariates involved in the corresponding

dependency.

3.3.6 Exogeneity

A variable is endogenous when there is a correlation between the parameter or variable

and the error term. endogeneity can arise as a result of measurement error,

autoregression with autocorrelated errors, simultaneity and omitted variables. Broadly,

http://en.wikipedia.org/wiki/Variable_(mathematics)

http://en.wikipedia.org/wiki/Correlation

http://en.wikipedia.org/wiki/Error_term

http://en.wikipedia.org/wiki/Measurement_error

http://en.wikipedia.org/wiki/Autoregression

http://en.wikipedia.org/wiki/Autocorrelation

http://en.wikipedia.org/wiki/Omitted-variable_bias


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a loop of causality between the independent and dependent variables of a model leads

to endogeneity.

3.3.6.1 Hausman Test

Here we use Hausman Test to test whether the factors are endogenous explanatory

variables or not. And we have:

H0: Factor is endogenous explanatory variables.

H1: Factor is endogenous explanatory variables.

A simple related test for endogeneity is the Hausman test. Under the null we consider

the model:

y = xβ + 𝜀

Under the alternative we consider the augmented model:

y = xβ + 𝛾��∗ + 𝜀∗

Where x∗are k* the explanatories under suspicion causing endogeneity approximated

by their estimates using the instruments.(k* is the number of explanatories which are

under consideration wrt endogeneity)

The idea is that under the null hypothesis of no endogeneity, x∗ represent irrelevant

additional variable, so γ=0.

Under the alternative the null model yields biased estimates. The test statistics is a

common F-test with K*and (n-K0-k*) degrees of freedom, where the restricted model

(under the null γ=0) is tested against the unrestricted (alternative) one. ( K0 is the

number of explanatories which are not under consideration wrt endogeneity) (Hahn,

Ham, and Moon, 2011).

Important to note: The test depends essentially on the choice of appropriate instruments.

http://en.wikipedia.org/wiki/Causality

http://en.wikipedia.org/wiki/Independent_variable

http://en.wikipedia.org/wiki/Dependent_variable


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Chapter 4: DATA ANALYSIS

In this section, we will be reviewing the descriptive statistic. Model specification test,

ordinary least square results and the diagnostic test were conducted to check whether

our model in-line with the OLS assumption and the Gauss–Markov theorem.

4.1 Descriptive Statistic

Table 4.1 below shows the descriptive statistics of each variable we have collected. But

the figures from different variables have a small divergence, as some of the variables

are transformed into natural logarithm form. As we can see the p-value from Jarque-

Bera test in the table, DEI and OP are normally distributed at the significant level of

5%.

Table 4.1: Summary Stats of Variables

Variable

s Mean

Media

n Mode Range

Std.Dev

.

Varianc

e

Skewnes

s

Kurtosi

s

JB.Stat

.

JB.P.Valu

e

SNP -

0.001

9

0.0143 -

0.314

9

0.489

1 0.0715 0.0051 -1.4524 8.3628 92.993 1.00E-03

MP -

5.91E-

04

0.0013 -0.043 0.058

5 0.0098

9.57E-

05 -1.8886 8.2779 105.31 1.00E-03

DEI 0.001

9 0.0019

0.001

8

1.16E-

04

3.43E-

05

1.17E-

09

-5.30E-

13 1.7993 3.604 0.0879

UI 0.009

1 0.0057

-

0.007

9

0.0528

0.0132 1.74E-

04 1.0693 3.4334 11.904 0.0111

UPR 2.400

2 2.18 1.83 4.19 0.971 0.9428 1.6589 4.9923 37.442 1.00E-03

UTS 2.938

5 3.432 -0.245 4.737 1.459 2.1288 -0.9326 2.4853 9.3588 0.0178

OP 5.371

6 5.378

5.0594

1.3986

0.3075 0.0946 -0.6609 3.4221 4.8128 0.055

EXP 0.002

2 0.0034

-

0.0133

0.026

1 0.0045

1.98E-

05 -1.5028 6.2741 49.383 1.00E-03

Figure 4.1 below shows the line graphs of each variable. As we can see some of the

variables have a trend pattern, which is violating the standard statistical assumptions

(e.g. CLRM assumptions). We need stationary data in the OLS model in order to obtain

more reliable results. Transformation of data is being used to solve this problem, we

tried first differencing on the data and realised minor trends could still be observed from

the graph (Figure 4.2), so we used second differencing of data to obtain stationary data,

as shown in Figure 4.3.


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Figure 4.1: Line Graph of Variables

Figure 4.2: Line Graph of First Differencing Variables


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Figure 4.3: Line Graph of Second Differencing Variables

4.2 Model Construction

After transforming the data to become stationary by using second differencing, we ran

an OLS regression with the second differencing dependent variable (ddSNP) against all

the other second differenced independent variables (Table 4.2). We found that ddDEI

and ddURP are highly insignificant, especially ddDEI with abnormal coefficient. We

decided to formulate a restricted model without these two independent variables, ddDEI

and ddURP, as shown in Table 4.3. As we can see from the result of the restricted

model, its adjusted R-square is higher than the unrestricted model.


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Table 4.2: OLS Regression Result of Unrestricted Model

Ordinary Least-squares Estimates

R-squared = 0.3405

Rbar-squared = 0.2482

sigma^2 = 0.0254

Durbin-Watson = 3.3708

Nobs, Nvars = 58, 8

***************************************************************

Variable Coefficient t-statistic t-probability

Cons -0.001362 -0.065003 0.948431

ddMP -2.961828 -2.120907 0.038911

ddDEI 2.27378E+13 1.025358 0.31013

ddUI -10.754665 -1.255579 0.215105

ddURP -0.020966 -0.319217 0.750892

ddUTS 0.096338 2.183662 0.033706

ddOP 0.267986 1.353807 0.181886

ddEXP 5.84429 2.02636 0.048079

Table 4.3: OLS Regression Result of Restricted Model


R-squared = 0.3252


sigma^2 = 0.0250


Nobs, Nvars = 58, 6

***************************************************************


Cons -0.001386 -0.06669 0.947085

ddMP -2.539738 -2.211562 0.031416

ddUI -9.919002 -1.383805 0.17233

ddUTS 0.105221 2.63457 0.011073

ddOP 0.223601 1.177789 0.244243

ddEXP 4.290565 1.754197 0.085288

The adjusted R-square of the restricted model is higher, and we suspect that the

unrestricted model has less explanation power for the data. But the result we get from

the Wald F-test (Table 4.4) shows that we have no evidence to reject the restrictions as

inconsistent with the data.


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Table 4.4: Wald F-test on Restricted and Unrestricted Model

Wald F-test

f-statistic f-probability

0.5796 0.5639

4.3 Preliminary Examination of Regression Residuals

By looking at the graph of residuals for the restricted and unrestricted model (Figure

4.4, and Figure 4.5), we can see that they are fluctuating around the mean of zero, we

assume that they are stationary.

Figure 4.4: Line Graph of the Residuals for Unrestricted Model


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Figure 4.5: Line Graph of the Residuals for Restricted Model

For the normality testing of residuals, we used the Jarque-Bera test on the residuals of

the restricted and unrestricted model. By looking at the histograms of the residuals, we

observe that it is not fully normal distributed. But the result from the Jarque-Bera test

shows that we have no evidence to reject the null hypothesis, which is the residuals are

normal distributed.


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Figure 4.6: Normality Test of the Residuals for Unrestricted Model

Jarque-Bera statistic = 4.1248 Jarque-Bera Probability = 0.0706

Figure 4.7: Normality Test of the Residuals for Restricted Model

Jarque-Bera statistic = 2.3255 Jarque-Bera Probability = 0.1834


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4.4 Model Specification

4.4.1 Information Criterions

In this section, we are using Akaike Information Criterion (AIC) and Schwarz

Information Criterion (SIC) to compare the restricted and unrestricted model. In Table

4.5, we can see that the AIC and the SIC value for the unrestricted Model is lower,

which shows that the Unrestricted Model is a better fitting model.

Table 4.5: Information Criterions

Unrestricted Model Restricted Model

AIC = -3.6208 SIC = -3.3764 AIC = -3.6645 SIC = -3.4900

4.4.2 Durbin Watson d Test

We are using Durbin Watson d Test to test for an omitted variable. Firstly, we try to

test the ddDEI variable, as it shows insignificant in our Unrestricted Model OLS result

and its coefficient is abnormally high. In this section, we try to see whether ddDEI is

captured by the residuals, when we removed it in our OLS regression. In Table 4.6, the

results show that we could not reject the null hypothesis at 1% significant level, it means

ddDEI do not get captured by the residuals, hence it is not an omitted variable, and we

decided to drop this variable.

Table 4.6: Test for ddDEI as Omitted Variable

Durbin Watson d Value = 2.3617

𝑑𝑙 = 1.382 𝑑𝑢 = 1.449 4 − 𝑑𝑢 = 2.5510 4 − 𝑑𝑙 = 2.6180

Besides this, we performed the same test on ddURP, as it shows highly insignificant in

the OLS regression as well. From Table 4.7, the result shows that we could not reject


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the null hypothesis at the 1% significant level, which means ddURP is not an omitted

variable, and we decided to drop this variable as well.

Table 4.7: Test for ddURP as Omitted Variable

Durbin Watson d Statistic = 2.0254

𝑑𝑙 = 1.382 𝑑𝑢 = 1.449 4 − 𝑑𝑢 = 2.5510 4 − 𝑑𝑙 = 2.6180

4.4.3 Ramsey RESET Test

In this part, we are testing the adequate of functional form for our model, by testing

linear against quadratic and cubic functional form. From the Table 4.8, we can see that

�� 2 is not significant, which mean quadratic functional form is not significant in our model.

And from the Table 4.9, we can see that the �� 3 is not significant as well, cubic functional form

is not significant in our model.

Table 4.8: Testing Against Quadratic Functional Form


R-squared = 0.3374


sigma^2 = 0.0250


Nobs, Nvars = 58, 7

***************************************************************


Cons -0.010817 -0.471317 0.639424

ddMP -2.097688 -1.696905 0.09581

ddUI -9.860284 -1.374773 0.175212

ddUTS 0.111505 2.754326 0.008132

ddOP 0.236281 1.24093 0.22031

ddEXP 3.857029 1.550377 0.127235

�� 2 0.883846 0.969647 0.336799


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Table 4.9: Testing Against Cubic Functional Form


R-squared = 0.3687


sigma^2 = 0.0243


Nobs, Nvars = 58, 8

***************************************************************


Cons -0.007205 -0.316823 0.752697

ddMP -1.035063 -0.742774 0.461094

ddUI -4.501316 -0.57347 0.568897

ddUTS 0.053503 0.984516 0.329602

ddOP 0.195311 1.030586 0.307695

ddEXP 3.366096 1.361456 0.179473

�� 2 0.422823 0.447318 0.656578

�� 3 6.324238 1.572559 0.122128

4.4.4 F-Test

By performing F-Test, we can see that the F Statistic is lesser than the critical value of

2.88334, as shown in Table 4.10. We can conclude that there is no problem in functional

form of our model.

Table 4.10: Result of F-Test

𝐹 = ((𝑅𝑛𝑒𝑤2 − 𝑅𝑜𝑙𝑑

2 )/𝑘)/(1 − 𝑅𝑛𝑒𝑤

2

𝑛 − 𝑝)

F Statistic = 1.3377 ; Critical Value = 2.88334

Where,

n = the sample size

k = the number of new regressors

p = the no. of parameters in the new model


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4.5 Description of Empirical Model

After we confirmed the model specification, we ended up with an empirical model as

follows:

ddSNP = 𝛼 + 𝛽(ddMP) + 𝛽(ddUI) + 𝛽(ddUTS) + 𝛽(ddOP) + 𝛽(ddEXP) + ε

ddSNP = S&P index (second differenced)

ddMP =Monthly Industrial Production (second differenced)

ddUI = Unexpected Inflation (second differenced)

ddUTS = Term Structure (second differenced)

ddOP = Oil Price (second differenced)

ddEXP = Consumption Expenditures (second differenced)

Furthermore, Table 4.11 below shows the results of OLS regression of the Empirical

model. We can see that ddMP, ddUI, and ddUTS is significant at 5% significant level;

in contrast, ddOP and ddEXP is not significant at 5% significant level. The R-squared

is 0.3253. This means that 32.52% of the variation of ddSNP can be explained by the

model.

Table 4.11: OLS Regression of Empirical Model


R-squared = 0.3252


sigma^2 = 0.0250


Nobs, Nvars = 58, 6

***************************************************************


Cons -0.001386 -0.06669 0.947085

ddMP -2.539738 -2.211562 0.031416

ddUI -9.919002 -1.383805 0.17233

ddUTS 0.105221 2.63457 0.011073

ddOP 0.223601 1.177789 0.244243

ddEXP 4.290565 1.754197 0.085288


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After we ran the OLS regression, we got the coefficients for each variable and we

substituted it into the Empirical Model, as follows:

ddSNP = −0.001386 + (−2.539738)(ddMP) + (−9.919002)(ddUI)

+ (0.105221)(ddUTS) + (0.223601)(ddOP) + (4.290565)(ddEXP)

+ ε

We can interpret this result as: when there is one unit increase in ddMP, ddSNP will

decrease by 2.54. While one unit decreases in ddUI, there will be increase of 10 units

in ddSNP, vice versa. ddSNP will be increased by 0.1053 units when ddUTS increase

by one unit. When ddOP increases by one unit, ddSNP increases by 0.2236 units. While

if ddEXP increases by one unit, ddSNP will increase by 4.3 units. All interpretations

here are assuming ceteris paribus.

4.6 Diagnostic Testing

4.6.1 Autocorrelation

To determine whether there is serial correlation of error term in our model, we have

used the Durbin-Watson test and Breusch-Godfrey Test.

4.6.1.1 Durbin-Watson Test

We obtained the Durbin-Watson Value from the OLS regression, the result shown in

Table 4.12. We have to reject the null hypothesis of the Durbin-Watson test. The

result shows that we have a negative autocorrelation problem in our model.

Table 4.12: Durbin-Watson Test

Durbin-Watson Value = 3.3956

𝑑𝑙 = 1.248 𝑑𝑢 = 1.598 4 − 𝑑𝑢 = 2.4020 4 − 𝑑𝑙 = 2.7520


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4.6.1.2 Breusch-Godfrey Test

From the result of Table 4.13, we can see that theε𝑡−1,ε𝑡−2 and ε𝑡−3 are significant at

1% level. We can conclude that there is serial autocorrelation in our model.

Table 4.13: Breusch-Godfrey Test


R-squared = 0.6675


sigma^2 = 0.0088


Nobs, Nvars = 58, 9

***************************************************************


Cons 0.002653 0.2147 0.830892

ddMP -0.780689 -1.13398 0.26232

ddUI 7.30608 1.614222 0.112901

ddUTS 0.019732 0.798857 0.428229

ddOP -0.099815 -0.869827 0.388636

ddEXP -2.269455 -1.525207 0.133637

ε𝑡−1 -1.107859 -8.764289 0

ε𝑡−2 -0.825458 -4.670847 0.000024

ε𝑡−3 -0.560972 -4.19637 0.000114

4.6.1.3 Remedy

Cochrane–Orcutt Estimation

We solve the autocorrelation problem by transforming the model to Generalised least

Squares, which is Cochrane-Orcutt AR(1) adjusted OLS regression as shown in Table

4.14. The result shows that the Durbin-Watson Statistic is closer to 2. Besides, the R-

squared is obviously higher, and the ddEXP is significant at 5% significant level in this

transformation model.


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Table 4.14: Cochrane–Orcutt Estimation

Cochrane-Orcutt serial correlation Estimates

R-squared = 0.4636


sigma^2 = 0.0119


Rho estimate = -0.7418

Rho t-statistic = -8.2781

Rho probability = 0.0000

Nobs, Nvars = 57, 6

***************************************************************

Iteration information

rho value convergence iteration

-0.700337 0.700337 1

-0.740943 0.040606 2

-0.741803 0.00086 3

-0.741819 0.000016 4

***************************************************************


Cons -0.00092 -0.110928 0.912109

ddMP -3.872029 -3.714257 0.000506

ddUI -13.932701 -3.137159 0.002832

ddUTS 0.072536 2.324864 0.024097

ddOP 0.243548 1.585996 0.118922

ddEXP 5.776322 2.383445 0.020913

4.6.2 Heteroskedasticity

4.6.2.1 Preliminary Examination of the Residuals

We test the normality of residuals using the following 2 methods:

Informal Method: Histogram and Q-Q Plot

Here we compare the residuals’ distribution of our regression model and residuals’

distribution of random normal. And find that residuals of both two regression models

follow normal distribution as shown in Figure 4.8.


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Figure 4.8: Graphs of the Residuals

OLS Residuals

OLS Residuals^2

Normal rand

Normal rand^2

From the Figure 4.9, we can see from the figure that, both of two lines lie approximately

on the line y=x. So, we conclude that residuals of both the regression models follow

normal distribution.

5 10 15 20 25 30 35 40 45 50 55-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

OLS

Resid

uals

5 10 15 20 25 30 35 40 45 50 550

0.05

0.1

0.15

0.2

0.25

OLS

Resid

uals

2

5 10 15 20 25 30 35 40 45 50 550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Norm

al ra

nd

2

5 10 15 20 25 30 35 40 45 50 55-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Norm

al ra

nd

-0.5 0 0.50

1

2

3

4

5

6

7

8OLS Residuals

-0.5 0 0.50

1

2

3

4

5

6

7Normal Random


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Figure 4.9: Q-Q Plot of Residuals

Formal method Jarque-Bera Test

Next, we perform the Jarque-Bera test on the normality on residuals, as shown in Table

4.15. If h =0, we can accept the null hypothesis that residuals of regression model follow

normal distribution. From the result, we can conclude that residuals follow normal

distribution at 1% significance level, at 5% significance level and at 10% significance

level respectively.

Table 4.15: Results of Jarque-Bera test

h jbstat p-value

Jarque Bera Test at 1% significance 0 2.3255 0.1834



-4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Standard Normal Quantiles

Qua

ntile

s of

Inp

ut S

ampl

eOLS Residuals

-4 -2 0 2 4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Standard Normal Quantiles

Qua

ntile

s of

Inp

ut S

ampl

e

Normal Random


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Goldfeld-Quandt Test

After confirming the residuals are in normal distribution, we proceed to the Goldfeld-

Quandt Test, as shown in Table 4.16. Since the p-value is larger than 0.05, we fail to

reject the null hypothesis, so the residuals in the population regression function are

homoscedastic.

Table 4.16: Result of Goldfeld-Quandt Test

Goldfeld-Quandt Test

f-stat f-prob Result

1.2999 0.2826 Homoscedastic

White’s General Heteroscedasticity Test

From the Table 4.17, we can see that the P-value is larger than 0.05, so we get the same

result: the residuals in the population regression function are homoscedastic.

Table 4.17: Result of White’s General Heteroskedasticity Test

White’s General Heteroskedasticity Test

Chisqr prob Result

14.0082 0.5509 Homoscedastic

Breusch Pagan Heteroscedasticity test

The result we got from this test shows the same result (Table 4.18), there is no

Heteroscedasticity in the residuals of the population regression function.


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Table 4.18: Result of Breusch Pagan Heteroscedasticity test

Breusch Pagan Heteroscedasticity test

Breush-Pagan LM-

statistic

Chi-squared

probability

Degrees of

freedom Result

3.38448744 0.6409 5 Homoscedastic

4.6.3 Mutlicollinearity

4.6.3.1 Preliminary Examination of Multicollinearity

In the first step, we analyze the data briefly. As we can see from the OLS result of the

Empirical Model, the R2 of our model is 0.3252, it is relatively low. That means the

possibility of having multicollinearity in our model is very low. Because there is

unlikely to be multicollinearity if we have high R2 but few significant t ratios. And from

the Figure 4.10, we can have an overview impression of the relationship between 5

variables and ddSNP. There is no significant multicollinearity problem.

Figure 4.10: Graph of Mutlicollinearity between Variables

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ddMP

ddS

NP

-0.01 -0.005 0 0.005 0.01 0.015-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ddUI

ddS

NP


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Multicollinearity is to check the relationship or correlation between independent

variables in the model. From Table 4.19, there is one pair of independent variables that

are highly correlated (>0.5), and the condition number is 0.2314, less than 1.

Table 4.19: Result of Correlation Test of Variables

ddMP ddUI ddUTS ddOP ddEXP

ddMP 1.0000 0.1418 0.0000 -0.0710 0.0409

ddUI 0.1418 1.0000 -0.2708 0.4988 0.2191

ddUTS 0.0000 -0.2708 1.0000 0.1064 0.0627

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ddUTS

ddS

NP

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ddOP

ddS

NP

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

ddEXP

ddS

NP

-0.01 -0.005 0 0.005 0.01 0.015-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

ddUI

ddO

P


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ddOP -0.0710 0.4988 0.1064 1.0000 0.2478

ddEXP 0.0409 0.2191 0.0627 0.2478 1.0000

Condition Number: 0.4731

4.6.3.2 Variance Inflation Factor (VIF) and Tolerance (TOL)

Therefore, as we can see in the Table 4.20, the estimation from VIF and TOL shows

that it has no serious Multicollinearity problem between OP and UI.

Table 4.20: Multivariate Variance Inflation Factors & Tolerance

VIF_ddMP 1.0594 TOL_ddMP 0.9439

VIF_ddDEI 1.6597 TOL_ddDEI 0.6025

VIF_ddUI 1.1967 TOL_ddUI 0.8356

VIF_ddOP 1.5330 TOL_ddURP 0.6523

VIF_ddEXP 1.0887 TOL_ddUTS 0.9185

VIF_ddOP VS ddUI 1.0887 TOL_ddOP VS ddUI 0.9185

4.6.3.3 Condition Indices (CI)

However, the estimation from CI is 18.5940 (Table 4.21), which shows that it has some

multicollinearity problems with our model. Although multicollinearity will deteriorate

statistical power; hypotheses testing may suffer from a type II error, we chose to do

nothing. Because it is a data deficiency problem and cannot be easily circumvented.


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Table 4.21: Belsley, Kuh, Welsch Variance-decomposition

4.6.4 Exogeneity

4.6.4.1 Hausman Test

We can see from the result (Table 4.22) that: input ddUI, ddUTS, ddOP and ddEXP are

endogenous variables.

Table 4.22: Result from Hausman Test

ddMP ddUI ddUTS ddOP ddEXP

Hausman stat 1.5792 7.2194 -4.4881 -4.9302 -4.9868

p-value 0.8691 1.7647e-09 0.0026 4.5855e-04 3.6119e-04

We can use instrument variable of relative endogenous variable to replace this

endogenous variable and get consistent but biased estimator. This instrument variable

must be correlated with the endogenous explanatory variables but cannot be correlated

with the error term in the explanatory equation. Besides, Two Stage Least Square

(TSLS) is a commom method, which includes create an investment variable to

eliminate endogeneity

K(x) CONS ddMP ddDEI ddUI ddOP ddEXP

1 1.00 0.00 0.00 0.00 0.00 0.00

2 0.00 0.00 0.00 0.83 0.00 0.00

7 0.00 0.00 0.00 0.01 0.66 0.00

54 0.00 0.94 0.00 0.00 0.00 0.00

117 0.00 0.01 0.00 0.00 0.05 0.98

346 0.00 0.05 1.00 0.15 0.29 0.02

CI = 18.5940


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CHAPTER 5: CONCLUSION

5.1 Discussions

From the results we presented in Chapter 4 Data Analysis, we can conclude that our

Empirical Model is not the best linear unbiased estimator. Since we performed the

preliminary test, our data has transformed to become stationary and the residuals are

normally distributed, we assume that our error term has zero population mean. In the

model specification test, we found that our Empirical model is correctly specified, and

have a correct functional form. Next, we found that we have a serial autocorrelation

problem in our model as well, and we use GLS regression to solve this problem.

Furthermore, we found that the error term of our model has constant variance, which is

homoscedasticity. Moreover, we found that our independent variables are correlated,

which leads to multicollinearity in our model, this will deteriorate the statistical power

of our model and hypotheses testing may suffer from a type II error. The

mutlicollinearity has little consequences to our Empirical Model’s result, so we chose

to do nothing about this problem. In the last diagnostic test, the result shows that some

of our independent variables are correlated with error terms, that means our model has

endogenous variables, therefore our OLS regression will yield biased and inconsistent

estimates as well as type II errors, because we have a mutlicollinearity problem.

5.2 Summary of Statistical Analyses

We solve the autocorrelation problem by transforming the model to Generalized least

Squares, which is Cochrane-Orcutt AR(1) adjusted OLS regression as shown in Table

5.1. Unfortunately we have mutlicollinearity and endogenous variables in our model.

This affects the reliability of our model, we assume this Generalised least Squares

method gave us the best results in this paper.


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Table 5.1: Result Cochrane–Orcutt Estimation

Cochrane-Orcutt serial correlation Estimates

R-squared = 0.4636


sigma^2 = 0.0119


Rho estimate = -0.7418

Rho t-statistic = -8.2781

Rho probability = 0.0000

Nobs, Nvars = 57, 6

***************************************************************

Iteration information

rho value convergence iteration

-0.700337 0.700337 1

-0.740943 0.040606 2

-0.741803 0.00086 3

-0.741819 0.000016 4

***************************************************************


Cons -0.00092 -0.110928 0.912109

ddMP -3.872029 -3.714257 0.000506

ddUI -13.932701 -3.137159 0.002832

ddUTS 0.072536 2.324864 0.024097

ddOP 0.243548 1.585996 0.118922

ddEXP 5.776322 2.383445 0.020913

From the Table 5.1, we can conclude that 46% of the variation of the dependent variable

(ddSNP) can be explained by the variation of the independent variables. There is only

one insignificant variable in our model, which is ddOP. While ddMP and ddUI are

significant at the 1% level, and they have a negative relationship with ddSNP. While

ddOP and ddEXP are significant at the 5% level, and have a positive relationship to the

ddSNP. When the variable increases by one unit, ddSNP will increase by 𝛽 value,

shown in Table 5.2.


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Table 5.2: Relationship between ddSNP with the independent variables

Variable 𝛽

ddMP -3.872029

ddUI -13.932701

ddUTS 0.072536

ddOP 0.243548

ddEXP 5.776322

5.3 Conclusion

The aim of this paper was to investigate the effect of macroeconomic determinants on

the performance of the S&P 500. As can be seen from our results, There is only one

insignificant variable in our model, which is Oil Prices (ddOP), Industrial production

(ddMP) and Unexpected inflation (ddUI), are significant at the 1% level, and they have

a negative relationship with the S&P500 Index (ddSNP), while Oil Price Changes

(ddOP) and Consumption Expenditure (ddEXP) are significant at the 5% level, and

have a positive relationship to the S&P500 Index. When the variable increases by one

unit, the S&P500 will increase by 𝛽 value, shown in Table 5.2. This tells us that if there

is a decrease in industrial production and unexpected inflation, there will be an increase

in the S&P500 index and vice, versa.

Our results are in line with that of Chen, Roll and Ross, 1986, especially regarding

Industrial production. Our results were also in line with Shanken and Weinstein (2006)

and Lamont (2000) regarding Industrial Production. However, our results were not

consistent with Cutler, Poterba and Summers, 1989 who found that Industrial

Production Growth is significantly positively correlated with real stock returns over the

period 1926 – 1986.They also found that Inflation, did not affect stock returns which

again, is not consistent with our results.

We conclude that Chen, Roll and Ross’s 1986 model is still valid and the results

conveyed by their research is not differ over time given the advances in technology and


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ease of access of information now, compared with back then. This result would give

investors in the market as well as governments more pertinent information regarding

macroeconomic factors that affect stock prices.

Our results might not be reliable, because it has some diagnostic problem and it is not

following Gauss-Markov theorem. But, in contrast, they are in line with several

previous literatures. Our suggestions for the future scholars are to solve the diagnostics

problem we are facing in this paper. Reexamining the same model in varies of stock

market could provide a more reliable and valid information to the participants in the

market.


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Appendices

%% Preliminary examinations of the Data

%% input variables % [Data,TXT,RAW]= xlsread('Data.xls',1,'A1:H60'); SNP = Data(:,1); MP = Data(:,2); DEI = Data(:,3); UI = Data(:,4); URP = Data(:,5); UTS = Data(:,6); OP = Data(:,7); EXP = Data(:,8); %% [1] Plot Data f = figure; subplot(4,2,1); plot(SNP); title('SNP'); subplot(4,2,2); plot(MP); title('MP'); subplot(4,2,3); plot(DEI); title('DEI'); subplot(4,2,4); plot(UI); title('UI'); subplot(4,2,5); plot(URP); title('URP'); subplot(4,2,6); plot(UTS); title('UTS'); subplot(4,2,7); plot(OP); title('OP'); subplot(4,2,8); plot(EXP); title('EXP');

% %% [1] Plot Data f = figure; subplot(4,2,1); plot(SNP); title('SNP'); subplot(4,2,2); plot(MP); title('MP'); subplot(4,2,3); plot(DEI); title('DEI'); subplot(4,2,4); plot(UI); title('UI'); subplot(4,2,5); plot(URP);


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title('URP');

subplot(4,2,6); plot(UTS); title('UTS'); subplot(4,2,7); plot(OP); title('OP'); subplot(4,2,8); plot(EXP); title('EXP'); %1st different D1 = LagOp({1,-1},'Lags',[0,1]); dSNP = filter(D1,SNP); dMP = filter(D1,MP); dDEI = filter(D1,DEI); dUI = filter(D1,UI); dURP = filter(D1,URP); dUTS = filter(D1,UTS); dOP = filter(D1,OP); dEXP = filter(D1,EXP); % plot 1st different f = figure; subplot(4,2,1); plot(dSNP); title('dSNP'); subplot(4,2,2); plot(dMP); title('dMP'); subplot(4,2,3); plot(dDEI); title('dDEI'); subplot(4,2,4); plot(dUI); title('dUI'); subplot(4,2,5); plot(dURP); title('dURP'); subplot(4,2,6); plot(dUTS); title('dUTS'); subplot(4,2,7); plot(dOP); title('dOP'); subplot(4,2,8); plot(dEXP); title('dEXP'); %2nd different D2 = D1*D1; ddSNP = filter(D2,SNP); ddMP = filter(D2,MP); ddDEI = filter(D2,DEI); ddUI = filter(D2,UI); ddURP = filter(D2,URP); ddUTS = filter(D2,UTS); ddOP = filter(D2,OP); ddEXP = filter(D2,EXP);

% plot 2nd different f = figure; subplot(4,2,1);


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plot(ddSNP); title('ddSNP'); subplot(4,2,2); plot(ddMP); title('ddMP'); subplot(4,2,3); plot(ddDEI); title('ddDEI'); subplot(4,2,4); plot(ddUI); title('ddUI'); subplot(4,2,5); plot(ddURP); title('ddURP'); subplot(4,2,6); plot(ddUTS); title('ddUTS'); subplot(4,2,7); plot(ddOP); title('ddOP'); subplot(4,2,8); plot(ddEXP); title('ddEXP'); %% [2] Measures of Central Tendancy Mean = mean(Data); Median = median(Data); Mode = mode(Data); % [3] Measures of Range Range = max(Data)-min(Data); Standdev = std(Data); Variance = var(Data); % [4] Higher 3rd and 4th Moments S = skewness(Data); K = kurtosis(Data); % [5] Perform a normality test [h,p,jbstat] = jbtest(SNP,.05); [h1,p1,jbstat1] = jbtest(MP,.05); [h2,p2,jbstat2] = jbtest(DEI,.05); [h3,p3,jbstat3] = jbtest(UI,.05); [h4,p4,jbstat4] = jbtest(URP,.05); [h5,p5,jbstat5] = jbtest(UTS,.05); [h6,p6,jbstat6] = jbtest(OP,.05); [h7,p7,jbstat7] = jbtest(EXP,.05); % [6] Restricted model X=[ones(length(ddSNP(:,1)),1),ddMP,ddUI,ddUTS,ddOP,ddEXP]; resultr=ols(ddSNP,X); prt(resultr) % [7] Unrestricted model X1=[ones(length(ddSNP(:,1)),1),ddMP,ddDEI,ddUI,ddURP,ddUTS,ddOP,ddEXP

]; resultu=ols(ddSNP,X1); prt(resultu) %

% % [8] Use an F-statistic to test the null that all the coefficients

are % % simultaneously zero. [fstat fprob] = waldf(resultr,resultu); disp('Wald F-test results'); [fstat fprob] % [9] Preliminary examination of Regression Residuals


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% plot(resultr.resid); ylabel('Residuals'); title('Restricted Model');

hist(resultr.resid); title('Restricted Model'); % plot(resultu.resid); ylabel('Residuals'); title('Unrestricted Model');

hist(resultu.resid); title('Unrestricted Model'); % % %% Normality of the residual % [h,p,jbstat] = jbtest(resultr.resid,.05)

[h,p,jbstat] = jbtest(resultu.resid,.05) % % % constant = ones(length(ddSNP),1); Data_Ori = [constant,ddMP,ddDEI,ddUI,ddURP,ddUTS,ddOP,ddEXP]; Data = [constant,ddMP,ddUI,ddUTS,ddOP,ddEXP];

% model specification % result_Ori = ols(ddSNP,Data_Ori); result = ols(ddSNP,Data);

% % SSR = sum(result_Ori.resid.^2); SSR1 = sum(result.resid.^2);

T = length(SNP); % % Information Criteria % Akaike = log(SSR/T)+(2*7)/T Schwarz = log(SSR/T)+(7*log(T))/T

Akaike1 = log(SSR1/T)+(2*5)/T Schwarz1 = log(SSR1/T)+(5*log(T))/T

%% % Durbin Watson d Test: a test for an omitted variable % Data2 = [constant,ddMP,ddUI,ddUTS,ddOP,ddEXP,ddURP]; result2 = ols(ddSNP,Data2);

Data3 = [ddDEI,result2.resid]; Data4 = sortrows(Data3,1); resid_sort = [Data4(:,2)]; % result3 = ols(resid_sort,constant); disp('Durbin Watson d Test: null of no autocorrelation (no omitted

variable)') result3.dw


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% %(n=60, k=1,1 percent significant) dl=1.382 du=1.449

4-du=2.5510 4-dl=2.6180 % % Data = [constant,ddMP,ddUI,ddUTS,ddOP,ddEXP]; result = ols(ddSNP,Data);

Data5 = [ddURP,result.resid]; Data6 = sortrows(Data5,1); resid_sort = [Data6(:,2)]; % result4 = ols(resid_sort,constant); disp('Durbin Watson d Test: null of no autocorrelation (no omitted

variable)') result4.dw % % Ramsey Reset Test: a test for adequate functional form % result = ols(ddSNP,Data);

predictsq = result.yhat.^2; predictcube = result.yhat.^3;

Data7 = [Data,predictsq]; Data8 = [Data,predictsq,predictcube];

result5 = ols(ddSNP,Data7); result5a = ols(ddSNP,Data8);

disp('Ramsey Reset Test: linear against quadratic functional form') prt(result5) %

disp('Ramsey Reset Test: linear against quadratic + cubic functional

form') prt(result5a) %quadratic + cubic functional form are not significant Fstat = ((result5a.rsqr - result.rsqr)/2)/((1-

result5a.rsqr)/length((SNP)-8)) % % Autocorrelation % Durbin-Watson Procedure % disp('Durbin-Watson Test: null of no autocorrelation') durbwat = result.dw

(k=7) dl=1.179 du=1.682

4-du=2.3180 4-dl=2.8210

(k=5)


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dl=1.248 du=1.598

4-du=2.4020 4-dl=2.7520 % % Breusch-Godfrey Test for serial correlation % plot(result.resid) hist(result.resid)

resid = result.resid; reslag1 = lag(resid,1); reslag2 = lag(resid,2); reslag3 = lag(resid,3); Data1 = [Data,reslag1,reslag2,reslag3]; result1 = ols(resid,Data1); prt(result1) % % Cochrane Orcutt procedure for AR(1) errors % % disp('Cochrane-Orcutt AR(1) adjusted OLS regression') results = olsc(ddSNP,Data) prt(results)

w% % heteroskedasticity

%Test the residuals for normality x = result.resid; y = normrnd(mean(x), std(x), length(x), 1); % figure; plot(x, 'Color', 'k') xlim([1, length(x)]) ylabel('OLS Residuals') % figure; plot(x.^2, 'Color', 'k') xlim([1, length(x)]) ylabel('OLS Residuals^2') % figure; plot(y, 'Color', 'k') xlim([1, length(y)]) ylabel('Normal rand') % figure; plot(y.^2, 'Color', 'k') xlim([1, length(y)]) ylabel('Normal rand^2') % Histogram figure; subplot(1,2,1) hist(x, 30) %histogram(x, 20) title('OLS Residuals') subplot(1,2,2) hist(y, 30) title('Normal Random')


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% Q-Q plot figure; subplot(1,2,1) qqplot(x) title('OLS Residuals') subplot(1,2,2) qqplot(y) title('Normal Random') % Jarque Bera test disp('Jarque Bera Test, null is normality of residuals') [H,P,JBSTAT] = jbtest(x, .05) disp('Jarque Bera Test, null is normality of residuals') [H,P,JBSTAT] = jbtest(x, .01) disp('Jarque Bera Test, null is normality of residuals') [H,P,JBSTAT] = jbtest(x, .10) %% % Goldfeld-Quandt B = sortrows(Data,2); % c = round(length(ddSNP)/5); n = round((length(ddSNP) - c)/2);

B1 = B(1:n, :); B2 = B(n+c:end, :); % Data1 = [ones(size(B1,1),1),B1(:,2:4)]; Data2 = [ones(size(B2,1),1),B2(:,2:4)]; % results1 = ols(B1(:,1), Data1); results2 = ols(B2(:,1), Data2); RSS1 = results1.resid'*results1.resid; RSS2 = results2.resid'*results2.resid; df1 = size(Data1, 1) - size(Data1, 2); df2 = size(Data2, 1) - size(Data2, 2); disp('Goldfeld Quandt F statistic, null is homoscedasticity and

normality of errors') fstat = (RSS2/df2)/(RSS1/df1) disp('Goldfeld Quandt F statistic, P value of F-statistic') fprob = fdis_prb(fstat, df1, df2) % 2. White heteroscedasticity test (does not assume normality of

errors) Data3 = [ones(length(ddSNP),1),ddMP,ddUI,ddUTS,ddOP,ddEXP]; results3 = ols(ddSNP,Data3); ressq3 = results3.resid.^2; % auxilary regression ddMP_sq = ddMP.*ddMP; ddUI_sq = ddUI.*ddUI; ddUTS_sq = ddUTS.*ddUTS; ddOP_sq = ddOP.*ddOP; ddEXP_sq = ddEXP.*ddEXP; ddMP_UI = ddMP.*ddUI;ddMP_UTS = ddMP.*ddUTS;ddMP_OP =

ddMP.*ddOP;ddMP_EXP = ddMP.*ddEXP; ddUI_UTS = ddUI.*ddUTS; ddUI_OP = ddUI.*ddOP;ddUI_EXP = ddUI.*ddEXP;% ddUTS_OP = ddUTS.*ddOP;ddUTS_EXP = ddUTS.*ddEXP; Data4 =

[ones(length(ddSNP),1),ddMP,ddUI,ddURP,ddUTS,ddOP,ddMP_UI,ddMP_UTS,dd

MP_OP,ddMP_EXP,ddUI_UTS,ddUI_OP,ddUI_EXP,ddUTS_OP,ddUTS_EXP]; results4 = ols(ressq3,Data4); % disp('White test, null is homoscedasticity') Chisqr = results4.rsqr*length(ressq3)


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prob = chis_prb(Chisqr,14) %3. Breusch Pagan heteroscedasticity test disp('Breusch Pagan Godfrey heteroscedasticity test, null is

homoscedasticity and normality of errors') bpagan(ddSNP,Data) % multicollinearity

% Unconditional Covariance and Correlation of Data matrix disp('Covariance and Correlation Matrices for Data') cov([ddMP ddUI ddUTS ddOP ddEXP]) corrcoef([ddMP ddUI ddUTS ddOP ddEXP]) R = corrcoef([ddMP ddUI ddUTS ddOP ddEXP]) cond(R) e = eig(R) e(5)/e(1) [V,D] = eig(R) %%% figure; subplot(1,2,1) plot(ddMP, ddSNP, '.b', 'MarkerSize', 22) xlabel('ddMP'); ylabel('ddSNP'); grid on; axis square;

subplot(1,2,2) plot(ddUI, ddSNP, '.r','MarkerSize', 22) xlabel('ddUI'); ylabel('ddSNP'); grid on; axis square;

figure; subplot(1,2,1) plot(ddUTS, ddSNP, '.k', 'MarkerSize', 22) xlabel('ddUTS'); ylabel('ddSNP'); grid on; axis square;

subplot(1,2,2) plot(ddOP, ddSNP, '.y', 'MarkerSize', 22) xlabel('ddOP'); ylabel('ddSNP'); grid on; axis square;

figure; subplot(1,2,1) plot(ddEXP, ddSNP, '.c', 'MarkerSize', 22) xlabel('ddEXP'); ylabel('ddSNP'); grid on; axis square; subplot(1,2,2) plot(ddUI, ddOP, '.m', 'MarkerSize', 22) xlabel('ddUI'); ylabel('ddOP'); grid on; axis square; % Variance Inflation Factors results_1 = ols(ddMP,[ones(length(ddUI),1),ddUI,ddUTS,ddOP,ddEXP]); results_2 = ols(ddUI,[ones(length(ddUI),1),ddMP,ddUTS,ddOP,ddEXP]); results_3 = ols(ddUTS,[ones(length(ddUI),1),ddUI,ddMP,ddOP,ddEXP]); results_4 = ols(ddOP,[ones(length(ddUI),1),ddUTS,ddMP,ddUI,ddEXP]); results_5 = ols(ddEXP,[ones(length(ddUI),1),ddUTS,ddMP,ddUI,ddOP]); results_6 = ols(ddOP,[ones(length(ddUI),1),ddUI]); % disp('Multivariate Variance Inflation Factors') disp('Multivariate Variance Inflation Factors MP') VIF1 = (1/(1-(results_1.rsqr))) disp('Multivariate Variance Inflation Factors DEI') VIF2 = (1/(1-(results_2.rsqr))) disp('Multivariate Variance Inflation Factors UI') VIF3 = (1/(1-(results_3.rsqr))) disp('Multivariate Variance Inflation Factors ERP') VIF4 = (1/(1-(results_4.rsqr))) disp('Multivariate Variance Inflation Factors UTS') VIF5 = (1/(1-(results_5.rsqr)))


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VIF6 = (1/(1-(results_5.rsqr))) %Tolerance Tol1 = 1./VIF1 Tol2 = 1./VIF2 Tol3 = 1./VIF3 Tol4 = 1./VIF4 Tol5 = 1./VIF5 Tol6 = 1./VIF6 % Condition Index disp('Condition Indices and associated statistics') bkw(Data) s = svd(Data); k = max(s)/min(s); ci = sqrt(k)

% exogeneiy

Data_exo1 =

[ones(length(ddMP),1),lag(ddMP,1),lag(ddMP,2),lag(ddMP,3)]; Result1 = ols(ddMP,Data_exo1); ddMP_hat = Result1.yhat; % 2. Second stage regression Data_exo2 = [ddMP_hat,ddUI,ddUTS,ddOP,ddEXP]; DataIV = [ones(length(ddSNP),1),Data_exo2]; ResultIV = ols(ddSNP, DataIV); prt(ResultIV) %if UI is endogeneous then its parameter estimate is biased. Data_exo1 =

[ones(length(ddUI),1),lag(ddUI,1),lag(ddUI,2),lag(ddUI,3)]; Result1 = ols(ddUI,Data_exo1); ddUI_hat = Result1.yhat; % 2. Second stage regression Data_exo2 = [ddUI_hat,ddUI,ddUTS,ddOP,ddEXP]; DataIV = [ones(length(ddSNP),1),Data_exo2]; ResultIV = ols(ddSNP, DataIV); prt(ResultIV) %if UTS is endogeneous then its parameter estimate is biased Data_exo1 =

[ones(length(ddUTS),1),lag(ddUTS,1),lag(ddUTS,2),lag(ddUTS,3)]; Result1 = ols(ddUTS,Data_exo1); ddUTS_hat = Result1.yhat; % 2. Second stage regression Data_exo2 = [ddUTS_hat,ddUI,ddUTS,ddOP,ddEXP]; DataIV = [ones(length(ddSNP),1),Data_exo2]; ResultIV = ols(ddSNP, DataIV); prt(ResultIV) %if OP is endogeneous then its parameter estimate is biased. Data_exo1 =

[ones(length(ddOP),1),lag(ddOP,1),lag(ddOP,2),lag(ddOP,3)]; Result1 = ols(ddOP,Data_exo1); ddOP_hat = Result1.yhat; % 2. Second stage regression Data_exo2 = [ddOP_hat,ddUI,ddUTS,ddOP,ddEXP]; DataIV = [ones(length(ddSNP),1),Data_exo2]; ResultIV = ols(ddSNP, DataIV); prt(ResultIV) %if EXP is endogeneous then its parameter estimate is biased. Data_exo1 =

[ones(length(ddEXP),1),lag(ddEXP,1),lag(ddEXP,2),lag(ddEXP,3)]; Result1 = ols(ddEXP,Data_exo1); ddEXP_hat = Result1.yhat; % 2. Second stage regression


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Data_exo2 = [ddEXP_hat,ddUI,ddUTS,ddOP,ddEXP]; DataIV = [ones(length(ddSNP),1),Data_exo2]; ResultIV = ols(ddSNP, DataIV); prt(ResultIV) % Hausman test beta_ols = ResultOLS.beta beta_iv = ResultIV.beta varbeta_ols = ResultOLS.sige*inv(DataOLS'*DataOLS); varbeta_iv =

ResultIV.sige*inv(DataIV'*DataIV); df = rank(varbeta_iv-varbeta_ols);

hausman = (beta_ols-beta_iv)'*inv(varbeta_iv-varbeta_ols)*(beta_ols-

beta_iv) pval = 1 - chis_prb(hausman.^2,df)

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