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ADVANCE ECONOMETRICS. LECTURE SLIDES, 2012 SPRING. The distinction between qualitative and quantitative data. - PowerPoint PPT PresentationTRANSCRIPT
LECTURE SLIDES, 2012 SPRING
ADVANCE ECONOMETRICS
The distinction between qualitative and quantitative dataThe microeconomist’s data on sales will have a
number corresponding to each firm surveyed (e.g. last month’s sales in the first company surveyed were £20,000). This is referred to as quantitative data.
The labor economist, when asking whether or not each surveyed employee belongs to a union, receives either a Yes or a No answer. These answers are referred to as qualitative data. Such data arise often in economics when choices are involved (e.g. the choice to buy or not buy a product, to take public transport or a private car, to join or not to join a club).
Economists will usually convert these qualitative answers into numeric data. For instance, the labor economist might set Yes = 1 and No = 0. Hence, Y1 = 1 means that the first individual surveyed does belong to a union, Y2 = 0 means that the second individual does not. When variables can take on only the values 0 or 1, they are referred to as dummy (or binary) variables.
What isDescriptive Research?
Involves gathering data that describe events and then organizes, tabulates, depicts, and describes the data.
Uses description as a tool to organize data into patterns that emerge during analysis.
Often uses visual aids such as graphs and charts to aid the reader
Descriptive Researchtakes a “what is” approach
What is the best way to provide access to computer equipment in schools?
Do teachers hold favorable attitudes toward using computers in schools?
What have been the reactions of school administrators to technological innovations in teaching?
Descriptive ResearchWe will want to see if a value or sample
comes from a known population. That is, if I were to give a new cancer treatment to a group of patients, I would want to know if their survival rate, for example, was different than the survival rate of those who do not receive the new treatment. What we are testing then is whether the sample patients who receive the new treatment come from the population we already know about (cancer patients without the treatment).
Economic and Econometric Models
The model of the economic behavior that has been derived from a theory is the economic model
After the unknown parameters have been estimated by using economic data foe the variables and by using an appropriate econometric estimation method, one has obtained an econometric model.
It is common to use Greek characters to denote the parameters.
C=f(Y) economic modelC=β1 + β2 Y , C=15.4+0.81Y econometric model
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Economic dataTime series data, historical dataCross sectional dataPanel dataVariables of an economic modelDependent variableIndependet variable, explanatory variable,
control variableThe nature of economic variables can be
endogeneous , exogeneous or lagged dependent
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A variable is an endogenous variable if the variable is determined in the model. Therefore dependant variable is always an endogenous variable
The exogenous variables are determined outside of the model.
Lagged dependent or lagged endogenous variables are predetermined in the model
The model is not necessary a model of only one equation if more equations have been specified to determine other endogenous variables of the system then the system is called a simultaneous equation model (SEM)
If the number of equation is identical to the number of endogenous variables, then that system of equation is called complete. A complete model can be solved for the endogenous variables.
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Static modelA change of an explanatory variable in period t is fully
reflected in the dependent variable in the same period t.Dynamic modelThe equation is called a dynamic model when lagged
variables have been specified.Structural equationsThe equations of the economic model as specified in the
economic theory, are called the structural equations.Reduced form modelA complete SEM can be solved for the endogenous
variables. The solution is called the reduced form model. The reduced form wil be used to stimulate a policy or to compute forecasts for the endogenous variables.
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Parameters and elasticities,The parameters are estimated by an estimator and
the result is called an estimateThe log transformation is a popular transformation
in econometric research, because it removes non-linearities to some extent.1
Stochastic termA disturbance term will be included at the right
hand side of the equation and is not observedAt the right hand side of the equation, two parts of
the specification ; the systematic part which concerns the specification of variables based on the economic theory; and the non-systematic part which is remaining random non-systematic variation.2
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Applied quantitative economic researchThe deterministic assumptions;It concerns the specification of the economic model, which
is formulation of the null hypothesis about the relationship between the economic variables of interest. The basic specification of the model originates from the economic theory. An important decision is made about the size of the model, whether one or more equation have to be specified.
the choice of which variables have to be included in the model stems from the economic theory
The availability and frequency of the data can influence the assumptions that have to be made
The mathematical form of the model has to be determined. Linear or nonlinear. Linear is more convenient to analyze.
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Evaluation of the estimation results,The evaluation concerns the verification of the validity
and reliability of all the assumptions that have been made
A first evaluation is obtained by using common sense and economic knowledge. This is followed by testing the stochastic assumptions by using a normality test, autocorrelation test, heteroscedasticity tests, etc. looking at a plot of the residuals can be very informative about cycles or outliers
If the stochastic assumptions have not been rejected , the deterministic assumptions can be tested by using statistical tests to test restrictions on parameters. The t-test and F-test can be performed. The coefficient of determination R2 can be interpreted.
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Hypothesis Testing
Purpose: make inferences about a population parameter by analyzing differences between observed sample statistics and the results one expects to obtain if some underlying assumption is true.
• Null hypothesis:• Alternative hypothesis: If the null hypothesis is rejected then the
alternative hypothesis is accepted.
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Hypothesis TestingA sample of 50 files from a file system is
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Confidence: 0.95
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Critical value =NORMINV(0.05,0,1)= -1.645.Region of non-rejection: Z ≥ -1.645.So, do not reject Ho. (Z exceeds critical value)
Steps in Hypothesis Testing1. State the null and alternative hypothesis.2. Choose the level of significance a.3. Choose the sample size n. Larger samples
allow us to detect even small differences between sample statistics and true population parameters. For a given a, increasing n decreases b.
4. Choose the appropriate statistical technique and test statistic to use (Z or t).
5. Determine the critical values that divide the regions of acceptance and nonacceptance.
6. Collect the data and compute the sample mean and the appropriate test statistic (e.g., Z).
7. If the test statistic falls in the non-reject region, Ho cannot be rejected. Else Ho is rejected.
Z test versus t test
1.Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s T-distribution.2. A T-test is appropriate when you are handling small samples (n < 30) while a Z-test is appropriate when you are handling moderate to large samples (n > 30).3. T-test is more adaptable than Z-test since Z-test will often require certain conditions to be reliable. Additionally, T-test has many methods that will suit any need.4. T-tests are more commonly used than Z-tests.5. Z-tests are preferred than T-tests when standard deviations are known.
One tail versus two tailwe were only looking at one “tail” of the distribution at
a time (either on the positive side above the mean or the negative side below the mean). With two-tail tests we will look for unlikely events on both sides of the mean (above and below) at the same time.
So, we have learned four critical values. 1-tail 2-tail = α .05 1.64 1.96, -1.96 = α .01 2.33 2.58/-2.58Notice that you have two critical values for a 2-tail
test, both positive and negative. You will have only one critical value for a one-tail test (which could be negative).
Which factors affect the accuracy of the estimate 1. Having more data points improves
accuracy of estimation.2. Having smaller errors improves
accuracy of estimation. Equivalently, if the SSR is small or the variance of the errors is small, the accuracy of the estimation will be improved.
3. Having a larger spread of values (i.e. a larger variance) of the explanatory variable (X) improves accuracy of estimation.
Calculating a confidence interval for β
If the confidence interval is small, it indicates accuracy. Conversely, a large confidence interval indicates great uncertainty over β’s true value. Confidence interval for β is
or
sb is the standard deviation of
Large values of sb will imply large uncertainty
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Nonparametric tests Parametric tests
Nominaldata
Ordinal data Ordinal, interval,ratio data
One group Chi squaregoodnessof fit
Wilcoxonsigned rank test
One group t-test
Twounrelatedgroups
Chi square Wilcoxon ranksum test,Mann-Whitneytest
Student’s t-test
Two relatedgroups
McNemar’stest
Wilcoxonsigned rank test
Paired Student’st-test
K-unrelatedgroups
Chi squaretest
Kruskal -Wallisone wayanalysis ofvariance
ANOVA
K-relatedgroups
Friedmanmatchedsamples
ANOVA withrepeatedmeasurements
Hypothesis testing involving R2: the F-statistic
R2 is a measure of how well the regression line fits the data or, equivalently, of the proportion of the variability in Y that can be explained by X. If R2 = 0 then X does not have any explanatory power for Y. The test of the hypothesis R2 = 0 can therefore be interpreted as a test of whether the regression explains anything at all
we use the P-value to decide what is “large” and what is “small” (i.e. whether R2 is significantly different from zero or not)
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F-TestUsage of the F-testWe use the F-test to evaluate hypotheses
that involved multiple parameters. Let’s use a simple setup:
Y = β0 + β1X1 + β2X2 + β3X3 + εi
F-TestFor example, if we wanted to know how
economic policy affects economic growth, we may include several policy instruments (balanced budgets, inflation, trade-openness, &c) and see if all of those policies are jointly significant. After all, our theories rarely tell
us which variable is important, but rather a broad category of variables.
F-statisticThe test is performed according to the
following strategy:1. If Significance F is less than 5% (i.e.
0.05), we conclude R2 ≠ 0.2. If Significance F is greater than 5%
(i.e. 0.05), we conclude R2 = 0.
Multiple regression modelThe model is interpreted to describe the conditional
expected wage of an individual given his gender, years of schooling and experience;
The coefficient β2 for malei measures the difference in expected wage between a male and a female with the same schooling and experience.
The coefficient β3 for schooli gives the expected wage difference between two individuals with the same experience and gender where one has additional year of schooling.
The coefficients in a multiple regression model can only be interpreted under a ceteris paribus condition.
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Estimation by OLS gives the following results;Dependent variable: wageVariable Estimate Standard Error t-ratioConstant -3.38 0.4650 -7.2692Male 1.3444 0.1077 12.4853School 0.6388 0.0328 19.4780Exper 0.1248 0.0238 5.2530s = 3.0462 R2 = 0.1326R2 = 0.13 F = 167.63The coefficient for malei suggest that if we compare an arbitrary male and
female with the same years of schooling and experience, the expected wage differential is 1.34$ with standard error 0.1077 which is statistically highly significant.
The null hypothesis that schooling has no effect on a persons wage, given gender and experience, can be tested using the t-test with a test statistic of 19.48.
The estimated wage increase from one additional year of schooling, keeping years of experience fixed is $0.64
The joint hypothesis that all three partial slope coefficients are zero, that is wages are not affected by gender, schooling or experience has to be rejected as well.
R2 is 0.1326 which means that the model is able explain 13.3% of the within sample variation in wages.
A joint test on the hypothesis that the two variables, schooling and experience both have zero coefficient by performing the F test.
With a 5% critical value of 3.0, the null hypothesis is obviously rejected. We can thus conclude that the model includes gender schooling and experience performs significantly better than model which only includes gender.
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Hypothesis testingWhen a hypothesis is statistically tested two types of errors can
be made. The first one is that we reject the null hypothesis while it is actually true- type I error. The second one is that the null hypothesis is not rejected while the alternative is true- type II error
The probability of a type I error is directly controlled by the researcher through his choice of the significance level α. When a test is performed at the 5% level, the probability of rejecting the null hypothesis while it is true is 5%.
The probability of a type II error depends upon the true parameter values. If the truth deviates much from the stated null hypothesis, the probability of such an error will be relatively small, while it will be quite large if the null hypothesis is close to the truth.
The probability of rejecting the null hypothesis when it is false, is known as the power of the test. It indicates how powerful a test is in finding deviations from the null hypothesis
Ordinary Least Squares (OLS)
Objective of OLS Minimize the sum of squared residuals:
where
Remember that OLS is not the only possible estimator of the βs.
But OLS is the best estimator under certain assumptions…
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CAPMExpected returns on individual assets are linearly
related to the expected return on the market portfolio. regression without an
intercept CAPM regression without interceptDependent variable: excess industry portfolio returnsIndustry food durables
constructionExcess market return 0.790 1.113 1.156
(0.028) (0.029)(0.025)
Adj.R2 0.601 0.741 0.804S 2.902 2.959 2.570
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CAPM regression with interceptDependent variable: excess industry portfolio returnsIndustry food durables
constructionConstant 0.339 0.064 -0.053
(1.128) (0.131) (0.114)Excess market return 0.783 1.111
1.157(0.028) (0.029) (0.025)
Adj.R2 0.598 0.739 0.803S 2.885 2.961 2.572
CAPM regression with intercept and January dummyDependent variable: excess industry portfolio returnsIndustry food durables constructionConstant 0.417 0.069 -0.094
(0.133) (0.137) (0.118)January dummy -0.956 -0.063 0.498
(0.456) (0.473) (0.411)Excess market return 0.788 1.112 1.155
(0.028) (0.029) (0.025)Adj.R2 0.601 0.739 0.804S 2.876 2.964 2.571
OLS AssumptionsBut OLS is the best estimator under certain
assumptions…Regression is linear in parameters2. Error term has zero population mean3. Error term is not correlated with X’s
(exogeneity)4. No serial correlation5. No heteroskedasticity6. No perfect multicollinearityand we usually add:7. Error term is normally distributed
ExogeneityAll explanatory variables are uncorrelated
with the error termE(εi|X1i,X2i,…, XKi,)=0Explanatory variables are determined outside
of the model (They are exogenous)What happens if assumption 3 is violated?Suppose we have the model,Yi =β0+ β1Xi+εi
Suppose Xi and εi are positively correlated When Xi is large, εi tends to be large as well.
ExogeneityWe estimate the relationship using the
following model:salesi= β0+β1pricei+εi
What’s the problem?
Assumption 3: ExogeneityWhat’s the problem?
What else determines sales of hamburgers?How would you decide between buying a
burger at McDonald’s ($0.89) or a burger at TGI Fridays ($9.99)?
Quality differssalesi= β0+β1pricei+εi quality isn’t an X
variable even though it should be.It becomes part of εi
Assumption 3: ExogeneityWhat’s the problem?
But price and quality are highly positively correlated
Therefore x and ε are also positively correlated.
This means that the estimate of β1will be too high
This is called “Omitted Variables Bias” (More in Chapter 6)
Serial CorrelationSerial Correlation: The error terms across
observations are correlated with each otheri.e. ε1 is correlated with ε2, etc.This is most important in time seriesIf errors are serially correlated, an increase in the
error term in one time period affects the error term in the next
Homoskedasticity: The error has a constant variance
This is what we want…as opposed toHeteroskedasticity: The variance of the error
depends on the values of Xs.
Perfect Multicollinearity
Two variables are perfectly collinear if one can be determined perfectly from the other (i.e. if you know the value of x, you can always find the value of z).
Example: If we regress income on age, and include both age in months and age in years.
Adjusted/Corrected R2
R2 = SSR/SST . As before, R2 measures the proportion of the sum of squares of deviations of Y that can be explained by the relationship we have fitted using the explanatory variables.
Note that adding regressors can never cause R2 to decrease, even if the regressors) do not seem to have a significant effect on the response of Y .
Adjusted (sometimes called \corrected") R2 takes into account the number of regressors included in the model; in effect, it penalizes us for adding in regressors that don't \contribute their part" to explaining the response variable.
Adjusted R2 is given by the following, where k is the number of regressors
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Interpreting Regression Results
β measures the expected change in Yi if Xi changes with one unit but all other variables in Xi do not change
In a multiple regression model single coefficients can only be interpreted under ceteris paribus conditions.
It is not possible to interpret a single coefficient in a regression model without knowing what the other variables in the model are.
The other variables in the model are called control variables
Economists are interested in elasticities rather than marginal effect. An elasticity measures the relative change in the dependant variable due to a relative change in one of the Xi variables.
iii XY
Linear model implies that elasticities are nonconstant and vary with Xi, while the loglinear model imposes constant elasticities.
Explaining log Yi rather than Yi may help reducing heteroscedasticity problems.
If Xi is a dummy variable (or another variable that may take negative values) we cannot take logarithm and we include the original variable in the model. Thus we estimate,
It is possible to include some explanatory variables in logs and some in levels. The interpretation of a coefficient β is the relative change in Yi due to absolute change of one unit in Xi. If Xi is a dummy variable for males β is the (ceteris paribus) relative wage differential between men and women.
iii XY log
Selecting regressorsWhat happens when a relevant variable is excluded from
the model and what happens when an irrelevant variable is included in the model.
Omitted variable biasIncluding irrelevant variables in your model, even though
they have a zero coefficient, will typically increase the variance of the estimators for the other model parameters. While including too few variables has the danger of biased estimates.
To find potentially relevant variables we can use economic theory.
General-to-spesific modelling approach – LSE methodology. This approach starts by estimating a general unrestricted model (GUM), which subsequently reduced in size and complexity by testing restrictions.
In presenting your estimation results, it is not a sin to have insignificant variables included in your specification
Be careful including many variables in your model can cause multicollinearity, in the end almost none of the variables appears individually significant
Another way to select a set of regressors, R2 measures the proportion of the sample variation in Yi , that is explained by variation in Xi.
If we were to extend the model by including Zi in the set of regressors the explained variation would never decrease, so that R2 will never decrease we include relevant additional variables in the model.
-not optimal because with too many variables we will not be able to say very much about the models coefficients. Becaause R2 does not punish the inclusion of many variables.
Better to use two of them. Trade-off between goodness of fir and the number of regressors employed in the model. Use adjusted R2
Adjusted R2 provides a trade-off between goodness of fit as measured by and simplicity of the model as measured by number of parameters
Akaikes Information CriterionSchwarz Bayesian Information CriterionModels with lower AIC and BIC are preferred.
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It is possible to test whether the increase in R2 is statistically significant.
Approproate F-statistics as follows, j: number of added
variables N-K : degrees of
freedom
If we want to drop two variables from the model at the same time we should be looking at a joint test( f or wald test) rather than at two separate t tests. Once the first variable is omitted from the model the second one may appear significant. This is of importance if collinearity exits between two variables.
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Comparing non-nested models
Both are interpreted as describing the conditional expectation of yi give xi and zi,
The two models are non-nested if zi includes a variable that is not in xi and vice versa.because both models are explaining the same endogenous variable, it is possible to use the adj R2, AIC and BIC criteria. An alternative and more formal way that can be used to compare the two models is that of encompassing.
- if a model A is believed to be correct model it must be able to encompass model B, that is, it must be able to explain model B s result. If model A is unable to do so, it has to be rejected.
There are two alternatives; non-nested F test and J test.
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Compared to non-nested F test, the J test involves only one restriction. This means that J test may be more powerful if the number of additional regressors in the non-nested F test is large. If the non-netsed F test involves only one additional regressor, ir is equivalent to the J test.
Two alternative ways that are nested is the choice between a linear and log linear functional form. Because the depeandant variables are different comparison of R2, AIC and BIC is inappropriate.
One way to test is Box-Cox transformation.
Misspecifying the functional form
Nonlinear models,Nonlinearity can arise in two ways; 1. model is
still linear in the parameters but non linear in its explanatory variables. We include nonlinear functions of xi as additional explanatory variables, for example the variables could be included in an individual wage equation. The resulting model is still linear in parameters and can still be estimated by OLS.
Nonlinear models can also be estimated by a nonlinear version of the LS method.
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Testing the Functional FormTo test whether the additional nonlinear terms in xi are
significant. This can be done using t-test, F-test or Wald testOr RESET test (regression equation specification error test)Testing for a structural breakThe coefficient in the model may be different before and
after a major change in macro-economic policy. The change in regression coefficient is referred to as a structural break.
A convenient way to express the general specification is given by;
Specification consist of two groups indicated by gi =0 and gi=1.
The null hypothesis is γ=0, in which case the model reduces to the restricted model
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A first way to test γ=0 is obtained by using the F-test;
K: number of regressors in the
restricted model (inc.intecept)
SUR and SR residual sum of squares of the unrestricted and restricted model
The above f test is referred to as the Chow test for structural change
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OLS results hedonic price functionDependant variable:log (price)Variable estimate standard error
t-ratioConstant 7.094 0.232
30.636Log(lot size) 0.400 0.028
14.397Bedrooms 0.078 0.015 5.017Bathrooms 0.216 0.023 9.386Air conditioning 0.212 0.024 8.923S= 0.2456 R2=0.5674 Adj R2=0.5642
F=177.41
OLS results hedonic price functionDependant variable:log (price) Variable estimate standard error
t-ratio Constant 7.745 0.216 35.801 Log(lot size) 0.303 0.027 11.356 Bedrooms 0.034 0.014 2.410 Bathrooms 0.166 0.020 8.154 Air conditioning 0.166 0.021
7.799 Driveway 0.110 0.028 3.904 Recreational room 0.058 0.026 2.225 Full basement 0.104 0.022 4.817 Gas for hot water 0.179 0.044 4.079 Garage places 0.048 0.011
4.178 Preferred area 0.132 0.023
5.816 Stories 0.092 0.013 7.268 S= 0.2104 R2=0.6865 Adj R2=0.6801 F=106.33
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OLS results hedonic price function Dependant variable:log (price) Variable estimate standard error
t-ratio Constant -4038.35 3409
-1.184 lot size 3.546 0.350 10.124 Bedrooms 1832 1047 1.75 Bathrooms 14335 1489 9.622 Air conditioning 12632 1555
8.124 Driveway 6687 2045 3.27 Recreational room 4511 1899 2.374 Full basement 5452 1588 3.433 Gas for hot water 12831 3217 3.988 Garage places 4244 840
5.050 Preferred area 9369 1669
5.614 Stories 6556 925 7.086 S= 15423 R2=0.6731 Adj R2=0.6664 F=99.97
HeteroscedasticityRemember Gaus-markov assumptions – BLUEThree ways of handling the problems of
heteroscedasticity and autocorrelation;Derivation of an altenative estimator that is best
linear unbiasedSticking to the OLS estimator but somehow
adjust the standard erors to allow heteroscedasticity or/and autocorrelation
Modify your model specification
Deriving an alternative estimator
We transform the model such that it satisfies the Gauss-markov conditions again
We obtain error terms that are homoscedastic and exhibit no autocorrelation
This estimator is referred to as the generalized least squares (GLS) estimator.
Hypothesis testing pg. 84-85
Testing heteroscedasticity,Testing equality of two unknown variances,
For one sided; Goldfeld-Quandt testThe Breusch-Pagan test – lagrange multiplier test
for heteroscedasticityThe white test – most common one but has limited
power against a large number of alternativesMultiplicative heteroscedasticity- has more power
but only against a limited number of alternatives.
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illustrationLabour: total employmentCapital: total fixed assetsWage: total wage costs divided by number of
workersOutput: value addedOLS result linear modelDependent variable: labourVariable estimate st.error t-ratioConstant 287.72 19.64 14.648Wage -6.742 0.501 -13.446Output 15.40 0.356 43.304Capital -4.590 0.269 -17.067S=156.26 R2=0.9352 Adj.R2=0.9348
F=2716.02
auxiliary regression Breusch-Pagan testDependent variable: ei
2
Variable estimate st.error t-ratioConstant -22719.51 11838.88 -1.919Wage 228.86 302.22 0.757Output 5362.21 214.35 25.015Capital -3543.51 162.12 -21.858S=94182 R2=0.5818 Adj.R2=0.5796
F=262.05High t-ratios and relatively high R2 indicate that the
error variance is unlikely to be constant.Breusch-Pagan test statistic = 569 (N) x R2=331.0As the asymtotic distribution under the null hypothesis
is a chi-squared with 3 degrees of freedom. This implies rejection of homoscedasticity.
Use logarithms to alleviate this problem
First step in handling the heteroscedasticity problem is to consider loglinear model,
If the production function is Cobb Douglas type Q=AKαLβ
OLS results loglinear modelDependent variable: log(labour)Variable estimate st.error t-ratioConstant 6.117 0.246 25.089logWage -0.928 0.071 -12.993logOutput 0.990 0.026 37.487logCapital -0.004 0.019 -0.197S=0.465 R2=0.8430 Adj.R2=0.8421
F=1011.02
A more general test is the White test. Run an auxiliary regression of squared OLS residuals upon all original regressors;
auxiliary regression Breusch-Pagan testDependent variable: ei
2
Variable estimate st.error t-ratioConstant 2.545 3.003 0.847Log(Wage) -1.299 1.753 -0.741log(Output) -0.904 0.560 -1.614Log(Capital) 1.142 0.376 3.039log2Wage 0.193 0.259 0.744log2Output 0.138 0.036 3.877log2Capital 0.090 0.014 6.401logWagelogoutput 0.138 0.163 0.849Logwagelogcapital -0.252 0.105 -2.399logoutputlogCapital -0.192 0.037 -5.197
S=0.851 R2=0.1029 Adj.R2=0.0884 F=7.12
Auxiliary regression multiplicative heteroscedasticityDependent variable: logei
2
Variable estimate st.error t-ratioConstant -3.254 1.185 -2.745logWage -0.061 0.344 -0.178logOutput 0.267 0.127 2.099logCapital -0.331 0.090 -3.659S=2.241 R2=0.0245 Adj.R2=0.0193
F=4.73F values leads to rejection of the null hypothesis of
homoscedasticity.
AutocorrelationWhen two or more consecutive error terms are
correlatedOLS remain unbiased but it becomes inefficient and
its standard errors are estimated in the wrong wayOccurs only when using time series dataFirst order autocorrelation
o error term is assumed to depend upon its predecessor as follows;
o o ut is an error term with mean zero and
canstant variance and exhibits no serial correlation
o |ρ|<1 first order autoregressive process is stationary(means, variance and covariances of εt do not change over time
ttt u 1
ttt u 1
Testing for first order autocorrelation
When ρ=0 no autocorrelation is present Breusch-Godfrey Lagrange multiplier test Durbin-Watson test dw ≈ 2-2ρ dw close to 2 indicates that first order
autocorrelation coefficient ρ is close to 0. if dw is much smaller than 2 this is an indication
for positive correlation
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What to do when you find A.correlationIn many cases finding of autocorrelation is
an indication that the model is misspecified. The most natural way is not to change your estimator from OLS to EGLS but to change your model.
Three types of misspecification may lead to a finding of autocorrelation in your OLS residuals:
Dynamic misspecificationOmitted variablesFunctional form misspecification
EndogeneitySome of explanatory variables can be correlated with
the error term, such that OLS estimator is biased and inconsistent
Autocorrelation with a lagged dependent variable;
If ρ≠0 OLS no longer yields consistent estimators. A possible solution is the use of maximum likelihood or instrumental variables
techniques Durbin-Watson test is not valid Use Breusch-Godfey lagrange multiplier test for autocorrelation, or Lagrange multiplier test
ttttt
ttt
tttt
yxy
yxy
11321
1
1321
Maximum likelihood
Instrumental Variable
SimultaneityWhen the model of interest contains behavioural
parameter, usually measuring the causal effects of changes in the explanatory variables, and one or more of these explanatory variables are jointly determined with the left hand side variable.
Illustration
The instrumental variables estimator
It is clear that people with more education have higher wages. It is less clear whether this positive correlation reflects a causal effect of schooling, or that individuals with a greater earnings capacity have chosen for more years of schooling. If the latter possibility is true, the OLS estimates on the return to schooling simply reflect differences in unobserved characteristics of working individuals and an increase in a persons schooling due to an exogenous shock will have no effect on this persons wage.
The problem of estimating the causal effect of schooling upon earnings
Most studies are based upon the human capital earnings function;
Wi denotes the log of individual earnings, Si denotes years of schooling and Ei denotes years of experience in the absense of information on actual experience, Ei sometimes replaced by potential experience measured as age-Si-6
Which variable serve as instrument. And instrument is thought of as a variable that affects the cost of schooling and thus the choice of schooling but not earnings. There is a long tradition of using family background variables, e.g. parents education as an instrument.
Table reports the results of an OLS regression of an individuals log hourlywage upon years of schooling, experience and experience squared and three dummy variables indicating whether the individual was black, lived in a metropolitan area and lived in the south
iiiii EESw 24321
Wage equation estimated by ols Dependent variable: log(wage) Variable estimate Standard error t-ratio Constant 4.7337 0.0676 70.022 Schooling 0.0740 0.0035 21.113 Exper 0.0836 0.0066 12.575 Exper2 -0.0022 0.0003 -7.050 Black -0.1896 0.0176 -10.758 Smsa 0.1614 0.0156 10.365 South -0.1249 0.0151 -8.259 S=0.374 R2=0.2905 Adj. R2=0.2891 F=204.93
Reduced form for schoolingestimated by ols Dependent variable: schooling Variable estimate Standard error t-
ratio Constant -1.8695 4.2984 -0.435 Age 1.0614 0.3014 3.522 age2 -0.0188 0.0052 -3.386 Black -1.4684 0.1154 -12.719 Smsa 0.8354 0.1093 7.647 South -0.4597 0.1024 -4.488 Lived near coll. 0.3471 0.1070 3.244 S=2.5158 R2=0.1185 Adj. R2=0.1168 F=67.29
Wage equation estimated by IV Dependent variable: log(wage) Variable estimate Standard error t-
ratio Constant 4.0656 0.6085 6.682 Schooling 0.1329 0.0514 2.588 Exper 0.0560 0.0260 2.153 Exper2 -0.0008 0.0013 -0.594 Black -0.1031 0.0774 -1.133 Smsa 0.1080 0.0050 2.171 South -0.0982 0.0288 -3.413 Instruments: age, age2, lived near college Used for exper, exper2 and schooling
OLS undeestimates the true causal effect of schooling
2SLS
GMMThis approach estimates the model parameters directly
from the moment conditions that are imposed by the model.
These conditions can be linear in the parameters but quite often are nonlinear.
The number of moment conditions should be at least as large as the number of unknown parameters
The great advantage of GMM is that1- it doesn’t require distributional assumption, like
normality2-it can allow for heteroscedasticity of unknown form3-it can estimate parameters efen if the model cannot
be solved analytically from the first order condition
Consider a linear model;With instrument vector zi the moment conditions
are
If εi is i.i.d. the optimal GMM is the instrumental variables estimator Consider the consumption based pricing model. Assume that there are risky
assets as well as riskless asset with certain return With one riskless asset and ten risky portfolios provide eleven moment
conditions with only two parameters to estimate These parameters can be estimated using the identity matrix as a suboptimal
weighting matrix, using the efficient two-step GMM estimatoror using iterated GMM estimator
Folowing Table presents the estimation results on the basis of the monthly returns using one step GMM and iterated GMM.
iii xy
0)( iiiii zxyEzE
GMM estimation results consumption based asset pricing model
One step GMM Iterated GMM Estimate s.e. Estimate s.e.δ 0.6996 (0.1436) 0.8273 (0.1162) γ 91.4097 (38.1178) 57.3992 (34.2203) ξ (df=9) 4.401 (p=0.88)5.685 (p=0.77)
The γ estimates are huge and rather imprecise. For the iterated GMM procedure, for example a 95% confidence interval for γ based on the approximate normal distribution is as large as (-9.67, 124.47). The estimated risk aversion coefficients of 57.4 and 91.4 are much higher than what is considered economically plausible.
To investigate the economic value of the model, it is possible to compute pricing errors
One can directly compute the average expected excess return according to the model, simply by replacing the population moments by the corresponding sample moments and using the estimated values for δ and γ. On the other hand the average excess returns on asset I can be computed from the data. (verbeek 157)
Maximum LikelihoodStronger than GMM approachIt assumes knowledge of the entire distribution, not just of
a number of momentsIf the distribution of a variable yi conditional upon a
number variables xi, is known up to a small number of unknown coefficients, we can use this to estimate these unknown parameters by choosing them in such a way that the resulting distribution corresponds as well as possible.
The conditional distribution of an observed phenomenon (the endogenous variable) is known, except for a finite number of unknown parameters.
These parameters will be estimated by taking those values for them that give the observed values the highest probability, the highest likelihood.
Specification TestsThe Wald, the likelihood ratio and Lagrange multiplier
principle.The wald test is generally applicable to any estimator
that is consistent and asymptotically normal.The likelihood ratio (LR) principle provides an easy
way to compare two alternative nested modelsLagrange multiplier (LM) test allow one to test
restrictions that are imposed in estimation. This makes the LM approach particularly suited for misspecification tests where a chosen specification of the model is tested for misspecification in several directions (like heteroscedasticity, non-normality or omitted variables)
Suppose that we have a sample of N=100 balls, of which 44% are red. If we test the hypothesis that p=0.5, we obtain wald, LR and LM test statistics of 1.46, 1.44 and 1.44 respectively. The 5% srtical value taken from the asymptotic chi-squared distribution with one degree of freedom is 3.84, so that the null hypothesis is not rejected at the 5% level with each of the three tests.
Testing for Omitted variables in non-linear models
HhhZi is a J-dimensional vector of explanatory
variables, independent of εi. The null hypothesis states H0:γ=0. note that under the assumptions above the F-test provides an exact test for γ=0 and there is no real need to look at asymptotic tests.
We can compute LM test for nonlinear models.
iiii zxy
Testing for heteroscedasticityThe squared OLS or maximum likelihood
residuals
Testing for autocorrelationAuxiliary regression of the OLS or ML.Quasi-maximum likelihood and moment
conditions testsConditional moment testInformation matrix testTesting for normality
Models with limited dependant variable
Binary Choice Models;To overcome the problems with the linear model,
there exist a class of binary choice models which are;
Probit modelLogit modelLinear probability model
An underlying latent modelIt is possible to derive a binary choice model from
underlying behavioural assumptions. This leads to a latent variable representation of the model.1
Because yi* is unobserved , it is referred to as a latent variable.
Our assumption is that an individual choices to work if the utility difference exceeds a certain threshold level, which can be set to zero without loss of generality.
Consequently, we observe yi=1 (job) if and only if yi*>0 and yi=0(no job) otherwise. Thus we have2
iii xy *
)(001 *iiiiiii xFxPxPyPyP
yi=1 if yi*>0yi=0 if yi*≤0Where εi independent of all xi. For the logit
model, the normal distribution is replaced bv the standard logictic one. Most commonly, the parameters in binary choice models (or limited dependent variable models in general) are estimated by the model of maximum likelihood.
)1,0(~,..* NIDxy iiii
Goodness of fitThere is no single measure for the goodness of fit in
binary choice models that contains only a constant as explanatory variable.
There are two goodness of fit measure defined in the lit.1
An alternative way to evaluate goodness of fit is comparing correct and incorrect predictions.2
Because it is possible that the model predicts worse than the simple model one can have wr1>wr0, in which case the Rp2 becomes negative. This is not good sign for the predictive quality of the model.
Illustration verbeek page; 197
0
12 1wr
wrRp
Specification tests in binary choice modelsThe likelihood function has to be correctly
specified. This means that we must be sure about the entire distribution that we impose on our data. Deviations will cause inconsistent estimators and in binary choice models this typically arises when the probability that yi=1 is misspecified as a function of xi. Usually such misspecifications are motivated from the latent variable model and reflect heteroscedasticity or non-normality of εi. In addition we may want to test for omitted variables without having to re-estimate the model. The most convenient framework for such tests is the lagrange multiplier framework.1
Multi response models1-ordered response models;
The probability that alternative j is chosen is the probability that the latent variable yi* is between two boundaries γj-1and γj . Assumng that εi is i.i.d. standard normal results in the ordered probit model. The logistic distribution gives the ordered logit model. For M=2 we are back at the binary choice model.
Illustration; married females, how much would you like to work? verbeek 203
Illustration; willingness to pay for natural areasMultinominal models; verbeek 208
jiji
iii
yifjy
xy
*
1
*
.....
Models for count dataIn certain applications, we would like to explain the
number of times a given event occurs, for example, how often a consumer visits a supermarket in a given week, or the number of patents a firm has obtained in a given year. Outcame might be zero for a subtantial part of the population. While the outcomes are discrete and ordered, there are two important differences with ordered response outcomes. First,he values of the outcome have a cardinal rather than just an ordinal meaning (4 is twice as much as 2 and two is twice as much as 1). Second, there is no natural upper bound to the outcomes
Poisson regression model. See page 211Illustration, see page 215 patent and Rand D
expenditures
Tobit ModelsThe dependant is continuous, but its range
may be constraint. Most commonly this occurs when the dependant variable is zero for a substantial part of the population but positive for the rest of the population.
Expenditures on durable goods, hours of work and the amount of foreign direct investment of a firm.
Estimation is usually done through maximum likelihood
Ilustration ; expenditures on alcohol and tobacco
Extensions of Tobit ModelsIt is coceivable that households with many children are
less likely to have positive expenditures, while if a holiday is taken up, the expected level of expenditures for such households is higher.
The tobit II model;Traditional model to describe sample selection problems
is the tobit II model also referred to as the sample selection model. In this context, it consists of a linear wage equation
where x1i denotes a vector of exogenous characteristics
(age, education, gender,…) and wi* denotes person i s wage. The wage wi* is not observed for people that are not working. To describe whether a person is working or not a second equation is specified, which is of the binary choice type. That is;
iii xh 222*
iii xw 111*
DdWhere we have the following observation rule:wi=wi*, hi=1 if hi*> 0Wi not obseved, hi=0 if hi*≤0Where wi denotes person i s actual wage. the binary
variable hi simply indicates working or not working. The model is completed by a distributional assumption on the unobserved erors (ε1i, ε 2i), usually a bivariate normal distribution with expectations zero, variances σ1
2, σ22 respectively,
and a covariance σ12 . The model is a standard probit model, describing the choice working or no working
Illustration: expenditures on Alcohol and Tobacco, verbeek page 233
iii xh 222*
Sample selection biasWhen the sample used in a statistical analysis is not
randomly drawn from a larger population, selection bias may occur.
If you interview people in the restaurant and ask how often they visit it, thoe that go there every day are much more likely to end up in the sample than those that visit every two weeks
Nonresponse may result in selection bias, people refuse to report their income are typicaly those with relatively high or low income levels.
Self selection of economic agents. Individuals select themselves into a certain state, working, union member, public sector employment, in a nonrandom way on the basis of economic arguments. In general who benefit most from being in a certain state will be more likely to be in this state.
Estimating treatment effectsAnother area where sample selection plays an
important role is in the estimation of treatment effects. A treatment effect refers to the impact of receiving a certain treatment upon a particular outcome variable, for example the effect of participating in a job training programme on future earnings. Because this effect may be different across individuals and selection into the training programme may be nonrandom.
The treatment effect is simply the coefficient for a treatment dummy variable in a regression model. Because interest is in the causal effect of the treatment, we need to worry about the potential endogeneity of the treatment dummy. We need to worry about selection into treatment.
Duration modelsTo explain the time it takes for an unemployed
person to find a job.It is Often used in labour market studies.Illustration; duration of the firm-bank
relationshipVerbeek page 250
Univariate time series models
Yt depends linearly on its previous value Yt-1 .that is,
Where εt denotes a serially uncorrelated innovation with a mean of zero and a constant variance. This process is called a first order autoregressive process or AR(1).
it says that the current value Yt equals a constant plus θ times its various value plus an unpredictable component εt
Assume that |θ|<1.The process for εt is an important building block of time
series models and it is referred to as white noise process. In this chapter it will always denote such s process that is homoscedastic and exhibits no autocorrelation
ttt YY 1
The expected value of Yt can be solved from
Assuming that E{Yt } does not depend on t, allow us to write
Writing time series models in terms of yt rather then Yt is often notationally more convenient.
Joint distribution of all values of Yt is characterized by the so-called autocovariances, the covariances between Yt and one of its lags, Yt-k
If we impose that variances and autocovariances do not depend on the index t. this is so called stationary assumption.
ttt
tt
t
yy
Yy
YE
1
,1
jtj
jtY
0
1 tt YEYE
Another simple time series model is the first order moving average process or MA(1) process, given by
This says that Y1 is a weighted average of ε1 and ε0 , Y2 is a weighted average of ε2 and ε1
The values of Yt are defined in terms of drawings from the white noise process εt
The simple moving average structure implies that observations that are two or more periods apart are uncorrelated.
this expression is referred to as the moving average representation of the autoregressive process
AR process is writen as an infinite order moving average processes. We can do so provided that |θ|<1.
For some purposes a moving average representation is more convenient than an autoregressive one.
1 tttY
jtj
jtY
0
We assume that the process for Yt is stationary.StationarityA stochastic process is said to be strictly stationary if
its properties are unaffected by a change of time origin
We will be only concerned with the means, variances and covariances of the series, and it is sufficient to impose that these moments are independent of time, rather than the entire distribution. This is referred to as weak stationarity or covariance stationarity.
Strict stationarity is stronger as it requires that the whole distribution is unaffected by a change in time horizon, not just the first and second order moments.
Autocovariance and autocorrelation1
Defining autocorrelation ρk as,
The autocorrelation considered as a function of k are referred to as autocorrelation function(ACF) or sometimes correlogram of the series Yt. The ACF plays a major role in modelling the dependencies among observations, because it characterizes the process describing the evolution of Yt over time.1
From the ACF we can infer the extent to which one value of the process is correlated with previous values and thus the length and strength of the memory of the process. It indicates how long and how strongly a shock in the process εt affects the values of Yt.
0
,cov
k
t
kttk YV
YY
For the AR(1) process,
we have
autocorrelation coefficients while for the MA(1)
processwe have
Consequently, a shock in an MA(1) process affects Yt in two periods only, while a shock in the AR(1) process affects all future observations with a decreasing affect.
Illustration. Verbeek page 260-261.
....4,3,2....0.1 21
1
1
kand
Y
YY
k
ttt
kk
ttt
General ARMA ProcessesWe define more general autoregressive and moving
average processesMA(q)
AR(p)
ARMA(p,q)
Often it is convenient to use the lag operator, denoted by L. it is defined by,
Most of the time lag operator can be manipulated just as if it were a constant;
So that, more generally.
qtqtttptpttt
tptpttt
qtqttt
yyyy
yyyy
y
......
...
...
22112211
2211
11
tt LyL )()(
1 tt yLy
212 )( tttt yLyLyLyL
pttp yyL
Using the lag operator allow us to write ARMA models in a concise way,
AR(1)
MA(1)
For more parsimonious representation, we may want to work with an ARMA model that contains both an AR and MA part. The general ARMA model can be written as
Lag polynomialsCommon roots, verbeek page 164-265
tjtj
jt
jtj
jt
j
j
jt
yy
Ly
10
00
)(
tt LyL )()(
Stationarity and Unit RootsStationarity of a stochastic process requires that the
variances and autocovariances are finite and independent of time. See Verbeek page 266, 267
A series which becomes stationary after first differencing is said to be integrated of order one, denoted I(1). If Δyt is described by a stationary ARMA(p,q) model, we say that yt is described by an autoregressive integrated moving average (ARIMA) model of order p, 1, q or in short an ARIMA (p,1,q) model.
First differencing quite often can transform a nonstationary series into a stationary one.
If a series must be differenced twice before it becomes stationary, then it is said to be integrated of order two, denoted I(2) and it must have two unit roots.
Testing for Unit RootsDickey-Fuller test, to test the null hypothesis that Ho: θ=1 (a unit root) , it is possible to use the standard t-statistic given by
ADF test: testing for unit roots in higher order autoregressive models1
KPSS test: the null hypothesis of trend stationary specifies that the variance of the random walk component is zero. The test is actually a lagrange multiplier test. t statistic is given by,
ILLUSTRATIONS, verbeek page 276
)(
1
seDF
2
1
2 ˆ/
T
ttSKPSS
ESTIMATION OF ARMA MODELSLeast SquaresMaximum Likelihood1CHOOSING A MODELBefore estimating any model it is common to
estimate autocorrelation and partial autocorrelation coefficients directly from the data. Often this gives some idea which model might be appropriate. After one or more models are estimated, their quality can be judged by checking whether the residuals are more or less white noise, and comparing them with alternative specifications. These comparisons can be based on statistical significance tests or the use of particular model selection criteria.
The Autocorrelation Function. ACFDescribes the correlation between Yt and its lag Yt-1 as a
function of k. The partial autocorrelation functionDIAGNOSTIC CHECKINGAs a last step in model building cycle some checks on the
model adequacy are required. Possibilities are doing a residual analysis and overfitting the specified model. For example if ARMA(p,q) model is chosen on the basis of the sample ACF and PACF, we could also estimate an ARMA(p+1,q) and an ARMA(p,q+1) model and test the significance of the additional parameters.
A residual analysis is usually based on the fact that the residuals of an adequate model should be approximately white noise. A plot of the residuals can be a useful tool in checking for outliers.
Moreover the estimated residual autocorrelation are usually examined. For A white noise series the autocorrelations are zero. Therefore the significance of the residual autocorrelations is often checked by comparing with approximate two standard error bounds . To check the overall acceptability of the residual autocorrelations, the Ljung-Box portmanteau test statistics is used;
o
If a model is rejected at this stage, the model building cycle has to be repeated.
T/2
2
1
12 k
K
kk r
kTTTQ
Criteria for model selectionAkaike s information Criterian (AIC)
Bayesian Information CriterianBoth criteria are likelihood based and
represent a different trade-off between fit as measured by the loglikelihood value, and parsimony, as measured by the number of free parameters, p+q+1.
Usually the model with the smallest AIC and BIC value is preferred.
ILLUSTRATION verbeek page 286.
TT
qpBIC
T
qpAIC
log1
ˆlog
12ˆlog
2
2
PREDICTING WITH ARMA MODELSA main goal of building a time series model is
predicting the future path of economic variables.The optimal predictorOur criterian for choosing a predictor from the many
possible ones is to minimize the expected quadratic prediction error.
Prediction accuracyIt is important to know how accurate this prediction isThe accuracy of the prediction decreases if we
predict further into the future.The MA(1) model gives more efficient predictors only
if one predicts one period ahead. More general ARMA models however will yield efficiency gains also in further ahead predictors.1
For the computation of the predictor, the autoregressive representation is most convenient.
The informational value contained in an AR(1) process slowly decays over time
For a random walk, with θ=1, the forecast error variance increase linearly with the forecast horizon.
In practical cases, the parameters in ARMA models will be unknown and we replace them by their estimated values. This introduces additional uncertainty in predictors. However this uncertainty is ignored.
ILLUSTRATION, the expectations theory of the term structure
AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY
In financial times series one often observes what is referred to as volatility clustering
In this case big shocks (residuals) tend to be followed by big shocks in either direction, and small shocks tend to follow small shocks.
ARCH: engel 1982, the variance of the error term at time t depends on the squared error terms from previous periods. The most simplest form is;
This specification does not imply that the process for εt is nonstationary. It just says that the squared values of
are correlated.
211
22 | tttt E
21
2 ,. tt and
The unconditional variance of εt is given by and has a stationary
solution
provided that 0≤α<1. Note that unconditional variance does not
depend on t
The ARCH(1) model is easily extended to an ARCH(p) processThe effect of a shock j periods ago on current volatility is
determined by the coefficient αj
In ARCH(p) model old shocks of more than p periods ago have no effect on current volatility.
k
21
2222
211
2
2
21
22
)(...
1
tptpttt
tt
L
EE
GARCH MODELSARCH models have been generalized in many different
waysGARCH(p,q) model can be written as;
or
Where α(L) and β(L) are lag polynomials. In practice a GARCH (1,1) specification often perform very well. It can be written as;
which has only three parameters to estimate. Non negative of requires that ϖ, α and β are non negative
21
21
2
2
1
2
1
2
)()(
ttt
jt
q
jjjt
p
jjt
LL
21
21
2 ttt
2t
If we define the surprise in the squared innovations as the GARCH(1,1) process can be written
as;
It implies that the effect of a shock on current volatility decreases over time. 1
An important restriction of the ARCH and GARCH specifications is their symmetry; only absolute values of the innovations matter, not their sign. That is a big negative shock has the same impact on future volatility as a big positive shock of the same magnitude.
An interesting extension is toward asymmetric volatility models, in which good news and bad news have a different impact on future volatility. Distinction between good and bad news is more sensible for stock markets than for exchange rates.
22ttt
12
12 )( tttt
An asymmetric model should allow for the possibility that an unexpected drop in price has a larger impact on future volatility than an expected increase in price of similar magnitude. A fruitful approach to capture such asymmetry is provided by nelson s exponential GARCH or EGARCH
Where α, β and γ are constant parameters. The EGARCH model is asymmetric as log as γ ≠0. when γ < 0 positive shocks generate less volatility than negative shocks
It is possible to extend the EGARCH model by including additional lags.
1
1
1
121
2 loglog
t
t
t
ttt
EstimationQuasi maximum likelihood estimationFeasible GLSIn financial markets, GARCH models are
frequently used to forecast volatility of returns, which is an important input for investment, option pricing, risk management and financial market regulation
ILLUSTRATION: Volatility in daily exchange rates verbeek page 303
Multivariate Time Series Models
Dynamic models with stationary variables let us consider two stationary variables Y1 and X1, and
assume that it holds that
We can think of Yt as company sales and Xt as advertising both in month t. if we assume that εt is a white noise process, independent of Xt and Yt values. The above relation referred to as an autoregressive distributed lag model. We use OLS to estimate it.
The interesting element in the model is that it describes the dynamic effects of a change in Xt upon current and future values of Yt. Taking partial derivatives, we can derive that the immediate response is given by
tttt XXYY 11011
0/ tt XY
This is referred to as the impact multiplier. An increase in X with one unit has an immediate impact on Y of units. The effect after one period is
And after two periods
This shows that after the first period, the effect is decreasing if |θ|<1. imposing this so-called stability condition allows us to determine the long run effect of a unit change in Xt. It is given by the long-run multiplier or equilibrium multiplier.
o
0
1011 // tttt XYXY
)(// 10112 tttt XYXY
1)...)(1(
.....)()(
1010
20
10100
This says that if advertising Xt increase with one unit, the expected cumulative increase in sales is given by
If the increase in Xt is permanent, the long run multiplier also has the interpretation of the expected long run permanent increase in Yt. An alternative way to formulate ARDL model is;
This formulation an example of an ECM. It says that the change in Yt due to current change in Xt, plus an error correction term. If Yt-1 is above the equilibrium value that corresponds to Xt-1, that is if the equilibrium error in square brackets is positive, an additional negative adjustments in Yt is generated. The speed of adjustment is determined by 1-θ, which is the adjustment parameter.(1-θ>0)
110
ttttt XYXY ])[1( 110
It is also possible to estimate ECM by OLS.Both the ARDL model and the ECM assume that
the values of Xt can be treated as given, that is , as being uncorrelated with the equations error terms. ARDL is describing the expected value of Yt given its own history and conditional upon current and lagged values of Xt. If Xt is simultaneously determined with Yt, OLS in ARDL and ECM would be inconsistent.
Partial adjustment model see verbeek page 312Generalized ARDL see verbeek 312
MODELS WITH NON-STATIONARY VARIABLESSpurious regressionConsider two variables, Yt and Xt, generated b two
independent random walks
o a researcher may want to estimate a regression model such as
The results from this regression are likely to be characterized by a fairly high R2 statistic, highly autocorrelated residuals and significant value of β. This is the well known problem of nonsense or spurious regression. Two independent nonstationary series are spuriously related due to the fact that they are both trending.
Including lagged variables in the regression is sufficient to solve many of the problems associated with the spurious regression.
ttt
ttt
ttt
XY
XX
YY
21
11
ttt
ttt
ttt
XY
XX
YY
21
11
COINTEGRATIONConsider two series integrated of order one, Yt and Xt,
and suppose that a linear relationship exist between them. This is reflected in the proposition that there exist some value β such that Yt- βXt is I(0), although Yt and Xt are both I(1). In such a cse it is said that Yt and Xt are cointegrated and they share common trend.
The presence of a cointegrating vector can be interpreted as the presence of a long run equilibrium relationship.
If Yt and Xt are cointegrated the error term is I(0). If not, error term will be I(1). Hence we can test for the presence of a cointegartion relationship by testing for a unit roo in the OL residuals. It seems that this can be done by using Dickey-Fuller test1
An alternative test for cointegration is basd on the usual Durbin watson statistic referred to as cointegrating regression durbin watson test or CRDW (page 316)
The presence of a cointegrating relationship implies that there exist a error correction model that desribes the short run dynamics consistently with the long run relationship.
COINTEGRATION AND ECMThe granger representation theorem states
that if a set of variables are cointegrated then there exist a valid error-correction representation of the data.
ttttt XYXY )( 1111
If Yt and xt are both I(1) but have a long run relationship, there must be some force which pulls he equilibrium error back towards zero.
The ECM describes how Yt and Xt behave in the short run consistent with a long run cointegrating relationship.
The repreenation theorem also holds conversely. If Yt and Xt ae both I(1) and have error correction representation, then they are necessarily cointegrated.
The concept of cointegration can be applied to nonstationary time series only.
ILUSTRATION; Long run purchasing power parity
VECTOR AUTOREGRESSIVE MODELSA VAR describes the dynamic evolution of a
number of variables from their common history. If we consider two variables, Yt and Xt, say the
VAR consists of two equations. A first order VAR would be given by;
Where ε1t and ε2t are two white noise processes (independent of the history of Y and X) that may be correlated. If, for example, θ12≠0 it means that the history of X helps explaining Y.
tttt
tttt
XYX
XYY
21221212
11121111
tttt
tttt
XYX
XYY
21221212
11121111
The VAR model implies univariate ARMA models for each of its comonents. The advantages of considering the components simultaneously that the model may be more parsimonious and includes fewer lags, and that more accurate forecasting is possible, because the information set is extended to also include the history of the other variables. Sims (1980) has advocated the use of VAR models instead of structural simultaneous equations models because the distinction between endogenous and exogenous variables does not have to be made a priori, and arbitrary constraints to ensure identification are not required.
We can use VAR model for forecasting in a straightforward way
To estimate A VAR we can simply use OLS equation by equation which is consistent because the white noise terms are assumed to be independent of the history of Yt.
Impulse Response FunctionIt measures the response of Yj,t+s to an impulse in Y1t,
keeping constant all other variables dated t and before.COINTEGRATION: The Multivariate CaseWe have a set of k I(1) variables, there may exist up to
k-1 independent linear relationships that are I(0). While any linear combination of these relationships is also I(0).
Vector error correction modelJohansen developed a maximum likelihood estimation
procedure, which allows one test for the number of cointegrating relations.
Trace test and maximum eigenvalue test. They are actually likelihood ratio tests.1
ILLUSTRATION: long run purchasing power parity, page 331
IllustrationMt: log of real M1 money balancesInflt: quarterly inflation rate (in % per year)Cprt: commercial paper rateyt: log real GDP (in billions of 1987 dollars)tbrt: treasury bill rateWe can think three possible cointegrating relationships
governing the long run behavior of these variablesFirst we can specify an equation for money demand as;
where β14 denotes the income elasticity. It can be expected that it is close to unity, corresponding to a unitary income elasticity., and β15 < 0.
tttt tbrym 115141
Second, if real interest rates are stationary we can expect that
Corresponds to a cointegrating relationship with β25 = 1. this is referred to as the Fisher relation, where are using actual inflation as a proxy for expected inflation.
Third, it can be expected that the risk premium, as measured by the difference between the commercial paper rate and the treasury bill rate, is stationary such that,
o with β35 =1
Before proceeding to the vector process of these five variables, let us consider the OLS estimates of the above three regressions.
ttt tbrl 2252inf
ttt tbrcpr 3353
ttt tbrcpr 3353
Univariate cointegrating regressions by OLS, intercept estimates not reported, standard errors in the parenthese Money Demand Fisher Equation Risk Prem.
mt: -1 0 0 Inflt: 0 -1 0 cprt: 0 0 -1 yt: 0.423 0 0
o (0.016) tbrt: -0.031 0.558 1.038 (0.002) (0.053)
(0.010) R2 0.815 0.409 0.984 dw 0.199 0.784 0.705 ADF(6) -3.164 -1.188 -3.975
Note that the OLS standard errors are inappropriate if the variables in the regression are integrated. Except for the risk premium equation, the R2 are not close to unity, which is an informal requirement for a cointegrating regression.
The empirical evidence for the existence of the suggested cointegration relationships between the five variables is somewhat mixed. Only for the risk premium equation we find and R2 close to unity, a sufficiently high dw statistic and a significant rejection of the ADF test for a unit root in the residuals. For the two other regressions there is little reason to reject the null hypotheses of no cointegration. Potentially this is caused by the lack of power of the tests that we employ, and it is possible that a multivariate vector analysis provides stronger evidence for the existence of cointegrating relationships between these five variables.
Plot of the residuals verbeek page 335The first step in the Johansen approach involves testing for
the cointegrating rank r. to compute these tests we need to choose the maximum lag length p in the vector autoregressive model. Choosing p too small will invalidate the tests and choosing p too large may result in a loss of power
Trace and maximum eigenvalue tests for cointegration
test statisticsNull hypothesis Alternative p=5 p=6 5%critical value λtrace- statisticH0: r=0 H1:r≥1 108.723 127.801
75.98H0: r≤1 H1:r≥2 59.18972.302 53.48
H0: r≤2 H1:r≥3 29.20135.169 34.87
H0: r≤3 H1:r≥4 13.78516.110 20.18 λmax- statistic
H0: r=0 H1:r=1 49.53455.499734.40
H0: r≤1 H1:r=2 29.98837.133 28.27
H0: r≤2 H1:r=3 15.41619.059 22.04
H0: r≤3 H1:r=4 9.637 11.860 15.87 See page 337-338
We impose two restrictions and, assuming that the money demand and risk premium relationships are the most likely candidates. We shall impoase that mt and cprt have coefficients of -1 ,0 and 0, -1, respectively. Economically we expect that inflt does not enter in any of the cointegrating vektors. With these two restrictions, the cointegrating vectors are ectimated by maximum likelihood, jointly with the coefficients in the vector error-correction model. The results for the cointegrating vectors are presented in table.
ML estimates of cointegrating vectors (after normalization) based on VAR with p=6 (standard errors in the parentheses)
Money Demand Risk Premium
mt: -1 0inflt: -0.023 0.041
(0.006)(0.031)
cprt: 0 -1yt: 0.425 -0.037 (0.033)
(0.173)tbrt: -0.028 1.017 (0.005) (0.026)Likelihood value: 808.2770
It is possible to test our a priori cointegrating vectors by using likelihood ratio tests. These tests require that the model is re estimated imposing some additional restrictions on the cointegrating vectors. This way we can test the following hypotheses,
The loglikelihood values for the complete model, estimated imposing respectively are given by 782.3459, 783.7761 and 782.3196. the likelihood ratio test statistics defined as twice the difference in loglikelihood values, for the three null hypothesis are thus given by 51.86,49.00 and 51.91. compared with the chi-squared critical values with 3,2 or 5 degrees of freedom, each of the hypotheses is clearly rejected.
1,0:
,1,0:
1,0:
25142422120
2524220
14120
c
b
a
H
H
H
cba andHHH 000 ,
As a last step we consider the VECM for this system. This corresponds to a VAR of order p-1=5 for the first differenced series, with the inclusion of two error correction terms in each equation. One for each cointegrating vector. The number of parameters estimated in this VECM is well above 100, so we shall concentrate on a limited part of the results only. The two error correction terms are given by
The adjustment coefficients in 5x2 matrix with their associated standard errors are reported in the table. The long run money demand equation contributes significantly to the short run movements of both money demand and income. The short run behaviour of money demand, inflation and the commercial paper rate appears to be significanatly affected by the long run risk premium relationship
687,0017.1037,0inf04,02
362.3028,0425.0inf023,01
ttttt
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tbrylcprecm
tbrylmecm
Estimated matrix of adjustment coefficients (standard errors in the parentheses) * indicates significance at the 5% level
Error correction termEquation ecm1t-1 em2t-2
Δmt 0.0276 0,0090* (0.0104) (0.0024)Δinflt 1.4629 -1.1618* (2.3210) (0.5287)Δcprt -2.1364 0.6626* (1.1494) (0.2618)Δyt 0.0687* 0,0013 (0.0121) (0.0028)Δtbrt -1.2876 0.3195 (1.0380) (0.2365)There is no statistical evidence that treasury bill rate
adjusts to any deviation from long run equlibria, so that it could be treated as weakly exogenous
Model based on panel data
An important adavaantage of the panel data compared to time series and cross-sectional data sets it allows identification of sertain paraameters or questions, without the need to make restrictive assumptions. For example panel data maake it possible to analyse changes on an individual level.
Consider a situation in wwhich the average consumption level rises with 2% from ona yeaar to another. Panel data can identfy whether this rise is a result of an increase of 2% for all individuals or an increase of 4% for approximately one half of the individuals and no change for the other half(or any other combinations)
Panel data are not only suitable to model or explain why individual units behave differently but also to model why a given unit behaves differently at different time periods.
We shall in the sequel index all variables by an I for the individual (i=1,2,3,…,N)and a t for the time period (t=1,2,3,…,T)
We can specify a linear model as, where βit measures the partial
effects of xit in period t for unit i.
Xit is a K-dimensional vector of explanatory variables, not including a constant. This means that the effects of a change in x are the same for all units and all periods, but that the average level for unit I may be different from that for unit j. the αi thus capture the effects of those variables that are peculiar to the i-th individual and that are constant over time. In the standard case, εit is assumed to be independent and identically distributed over individuals and time, with mean 0 and variance . If we treat the αi as N fixed unknown parameters the model is referred to as the standard fixed effects model
itititit xy
2
itij
N
jjit xdy
1
An alternative approach assumes that the intercepts of the individuals are different but that they can be treated as drawings from a distribution with mean μ and variance . The essential assumption here is that these drawings are independent of the explanatory variables in xit. This leads to random effect model where the individual effects αi are treated as random
Most panel data models are estimated under either the fixed effects or the random effects assumption
2
Efficiency of Parameter EstimationBecause panel data sets are typically larger
than cross-sectional or time series data sets, and explanatory variables vary over two dimensions (individuals and time) rather than one, estimators based on panel dat are quite often more accurate tha from other sources. Even with identical sample sizes the use of a panel data set wil often yield more efficient estimators than a series of independent cross sections (where different units are sampled in each period)1
Identification of parametersA second adavantage of the availability of
panel dat is that it reduces identification problem.
Estimating the model under fixed effects assumption eliminates αi from the error term and, consequently, eliminates all endogeneity problems relating to it. 1
THE STATIC LINEAR MODEL1. Fixed effects modelThis is simply a linear regression model in which the
intercept terms vary over the individual units i,
Where it is usually assumed that all xit are independent of all εit. We can write this in the usual regression framework by including a dummy variable for each unit i in the model
we thus have a set of N dummy variables in the model. The parameters can be estimated by OLS. The implied estimator for β is referred to as the least squares dummy variable estimator (LSDV)
ititiit xy
ititij
N
jjit xdy
1
We eliminate the individual effect of αi first by transforming the data
This is a regression model in deviations from individual means and does not include the individual effects αi. The transformation that produces observations in deviation from individual means is called the within transformation. The OLS estimator for β obtained from this transformed model is often called the within estimator or fixed effects estimator, and it is exactly identical to the LSDV estimator.
We call xit strictly exogenous. A strictly exogeneous variable is not allowed to depend on current, future and past values of the error term.
The fixed effects model concentrates on differences within individuals. It is explaining to what extent yit differs from yi and does not explain why yi is different from yj
)()( iitiitiit xxyy
The random effect modelIt is commonly assumed in regresion analysis that all
factors that affect the dependent variable, but that have not been included as regressors, can be appropriately summarized by a random error term. In our case this leads to the assumption that the αi are random factors, independently and identically distributed over individuals. Thus write the random affects model as;
Where is treated as an error term consisting of two components: and individual specific component, which does not vary over time, and the remainder component, which is assumed to be uncorrelated over time.
That is all correlation of the error terms over time is attributed to the individual affects of xjs (for all j and s). This implies that OLS estimator is unbiased and consistent.
iti itiitit xy
iti
Estimators for the parameter vector β;1- the between estimator, exploiting the between dimension
of the data (differences between individuals) determined as the OLS in a regression of individual averages of y on individual averages of x. the explanatory variables are strictly exogenous and uncorrelated with the individual specific affect αi.
2- the fixed effect estimator, exploiting the within dimension of the data (differences between individuals) determined as the OLS estimator in a regression in deviations from individual means.
3-the OLS estimator, exploiting both dimensions (within and between) but not efficiently. The explanatory variables to be uncorrelated with αi but does not impose that they are strictly exogenous.
4- The random effect (EGLS) estimator, combining the information from the between and within dimensions in an efficient way.
Fixed effects or random effectsThe fixed effects approach is conditional upon the values
for αi.If the individuals in the sample are one of a kind and
cannot be viewed as a random draw some underlying population. When I denotes countries, large companies or industries, and predictions we want to make are for a particular country, company or industry.
In contrast the random effects approach is not conditional upon the individual αis, but integrates them out. In this case, we are usually not interested in the particular value of some persons αi, we just focus on arbitrary individuals that have certain characteristics. The random effects approach allows one to make inference with respect to the population characteristics.
Hausman test: tests whether the fixed effects and random effects estimator are significantly different.
Goodness of fitThe computation of goodness of fit measures in
panel data applications is somewhat uncommon. One reason is the fact that one may attach different importance to explaining the within and between variation in the data. Another reason is that the usual R2 or adjusted R2 criteria are only appropriate if the model is estimated by OLS.
The goodness of fit measures are not adequate to choose between alternative estimators
Robust Inference, page 355Testing for heteroscedasticity and
autocorrelatio pg.357ILLUSTRATION- 358 . Explaining individual
wages
Dynamic linear models
An autoregressive panel data model
Dynamic models with exogenous variables
Illustration- wage elasticities of labour demand
Non-stationarity, unit roots and cointegration
Panel data unit root test
Panel data cointegration tests
Models with limited dependant variables
Binary choice models
The fixed effects logit model
The random affects probit model
Tobit models
Incomplete panels and selection bias