chapter 9 regression with time series data: stationary variables

Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data:

Stationary Variables

Chapter 9Regression with Time Series

Data:Stationary Variables

Walter R. Paczkowski Rutgers University



9.1 Introduction9.2 Finite Distributed Lags9.3 Serial Correlation9.4 Other Tests for Serially Correlated Errors9.5 Estimation with Serially Correlated Errors9.6 Autoregressive Distributed Lag Models9.7 Forecasting9.8 Multiplier Analysis

Chapter Contents



9.1 Introduction



When modeling relationships between variables, the nature of the data that have been collected has an important bearing on the appropriate choice of an econometric model– Two features of time-series data to consider:

1. Time-series observations on a given economic unit, observed over a number of time periods, are likely to be correlated

2. Time-series data have a natural ordering according to time

9.1Introduction



There is also the possible existence of dynamic relationships between variables – A dynamic relationship is one in which the

change in a variable now has an impact on that same variable, or other variables, in one or more future time periods

– These effects do not occur instantaneously but are spread, or distributed, over future time periods

9.1Introduction



9.1Introduction FIGURE 9.1 The distributed lag effect



Ways to model the dynamic relationship:1. Specify that a dependent variable y is a

function of current and past values of an explanatory variable x

• Because of the existence of these lagged effects, Eq. 9.1 is called a distributed lag model

9.1Introduction

9.1.1Dynamic Nature of Relationships

1 2( , , ,...)t t t ty f x x x Eq. 9.1



Ways to model the dynamic relationship (Continued):2. Capturing the dynamic characteristics of time-series

by specifying a model with a lagged dependent variable as one of the explanatory variables

• Or have:

–Such models are called autoregressive distributed lag (ARDL) models, with ‘‘autoregressive’’ meaning a regression of yt on its own lag or lags

9.1Introduction


Eq. 9.2 1( , )t t ty f y x

Eq. 9.3 1 1 2( , , , )t t t t ty f y x x x



Ways to model the dynamic relationship (Continued):3. Model the continuing impact of change over

several periods via the error term

• In this case et is correlated with et - 1

• We say the errors are serially correlated or autocorrelated

9.1Introduction


Eq. 9.4 1( ) ( )t t t t ty f x e e f e



The primary assumption is Assumption MR4:

• For time series, this is written as:

– The dynamic models in Eqs. 9.2, 9.3 and 9.4 imply correlation between yt and yt - 1 or et and et

- 1 or both, so they clearly violate assumption MR4

9.1Introduction

9.1.2Least Squares Assumptions

cov , cov , 0 for i j i jy y e e i j

cov , cov , 0 for t s t sy y e e t s



A stationary variable is one that is not explosive, nor trending, and nor wandering aimlessly without returning to its mean

9.1Introduction

9.1.2aStationarity



9.1Introduction

9.1.2aStationarity

FIGURE 9.2 (a) Time series of a stationary variable



9.1Introduction

9.1.2aStationarity

FIGURE 9.2 (b) time series of a nonstationary variable that is ‘‘slow-turning’’ or ‘‘wandering’’



9.1Introduction

9.1.2aStationarity

FIGURE 9.2 (c) time series of a nonstationary variable that ‘‘trends”



9.1Introduction

9.1.3Alternative

Paths Through the Chapter

FIGURE 9.3 (a) Alternative paths through the chapter starting with finite distributed lags



9.1Introduction

9.1.3Alternative

Paths Through the Chapter

FIGURE 9.3 (b) Alternative paths through the chapter starting with serial correlation



9.2 Finite Distributed Lags



Consider a linear model in which, after q time periods, changes in x no longer have an impact on y

– Note the notation change: βs is used to denote the coefficient of xt-s and α is introduced to denote the intercept

0 1 1 2 2t t t t q t q ty x x x x e Eq. 9.5

9.2Finite

Distributed Lags



Model 9.5 has two uses:– Forecasting

– Policy analysis• What is the effect of a change in x on y?

1 0 1 1 2 1 1 1T T T T q T q Ty x x x x e Eq. 9.6

( ) ( )t t ss

t s t

E y E yx x

Eq. 9.7

9.2Finite

Distributed Lags



Assume xt is increased by one unit and then maintained at its new level in subsequent periods – The immediate impact will be β0 – the total effect in period t + 1 will be β0 + β1, in

period t + 2 it will be β0 + β1 + β2, and so on • These quantities are called interim

multipliers– The total multiplier is the final effect on y of

the sustained increase after q or more periods have elapsed

0

βq

ss

9.2Finite

Distributed Lags



The effect of a one-unit change in xt is distributed over the current and next q periods, from which we get the term ‘‘distributed lag model’’– It is called a finite distributed lag model of order q • It is assumed that after a finite number of

periods q, changes in x no longer have an impact on y

– The coefficient βs is called a distributed-lag weight or an s-period delay multiplier

– The coefficient β0 (s = 0) is called the impact multiplier

9.2Finite

Distributed Lags



9.2Finite

Distributed Lags

9.2.1Assumptions

TSMR1.TSMR2. y and x are stationary random variables, and et is independent of current, past and future values of x.TSMR3. E(et) = 0TSMR4. var(et) = σ2

TSMR5. cov(et, es) = 0 t ≠ sTSMR6. et ~ N(0, σ2)

0 1 1 2 2β β β β , 1, ,t t t t q t q ty x x x x e t q T

ASSUMPTIONS OF THE DISTRIBUTED LAG MODEL



Consider Okun’s Law– In this model the change in the unemployment rate

from one period to the next depends on the rate of growth of output in the economy:

–We can rewrite this as:

where DU = ΔU = Ut - Ut-1, β0 = -γ, and α = γGN

9.2Finite

Distributed Lags

9.2.2An Example: Okun’s Law

1t t t NU U G G Eq. 9.8

0βt t tDU G e Eq. 9.9



We can expand this to include lags:

We can calculate the growth in output, G, as:

9.2Finite

Distributed Lags


Eq. 9.100 1 1 2 2β β β βt t t t q t q tDU G G G G e

Eq. 9.111

1

100t tt

t

GDP GDPGGDP



9.2Finite

Distributed Lags


FIGURE 9.4 (a) Time series for the change in the U.S. unemployment rate: 1985Q3 to 2009Q3



9.2Finite

Distributed Lags


FIGURE 9.4 (b) Time series for U.S. GDP growth: 1985Q2 to 2009Q3



9.2Finite

Distributed Lags


Table 9.1 Spreadsheet of Observations for Distributed Lag Model



9.2Finite

Distributed Lags


Table 9.2 Estimates for Okun’s Law Finite Distributed Lag Model



9.3 Serial Correlation



When is assumption TSMR5, cov(et, es) = 0 for t ≠ s likely to be violated, and how do we assess its validity? –When a variable exhibits correlation over time,

we say it is autocorrelated or serially correlated• These terms are used interchangeably

9.3Serial

Correlation



9.3Serial

Correlation

9.3.1Serial

Correlation in Output Growth

FIGURE 9.5 Scatter diagram for Gt and Gt-1



Recall that the population correlation between two variables x and y is given by:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

cov ,ρ

var varxy

x y

x y



For the Okun’s Law problem, we have:

The notation ρ1 is used to denote the population correlation between observations that are one period apart in time– This is known also as the population autocorrelation of

order one. – The second equality in Eq. 9.12 holds because

var(Gt) = var(Gt-1) , a property of time series that are stationary

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

1 11

1

cov , cov ,ρ

varvar vart t t t

tt t

G G G GGG G

Eq. 9.12



The first-order sample autocorrelation for G is obtained from Eq. 9.12 using the estimates:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

1 12

2

1

1cov ,1

1var1

T

t t t tt

T

t tt

G G G G G GT

G G GT



Making the substitutions, we get:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

Eq. 9.13

12

1 2

1

T

t tt

T

tt

G G G Gr

G G



More generally, the k-th order sample autocorrelation for a series y that gives the correlation between observations that are k periods apart is:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

Eq. 9.14

1

2

1

T

t t kt k

k T

tt

y y y yr

y y



Because (T - k) observations are used to compute the numerator and T observations are used to compute the denominator, an alternative that leads to larger estimates in finite samples is:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

Eq. 9.15

1

2

1

1

1

T

t t kt k

k T

tt

y y y yT kr

y yT



Applying this to our problem, we get for the first four autocorrelations:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

Eq. 9.16 1 2 3 40.494 0.411 0.154 0.200r r r r



How do we test whether an autocorrelation is significantly different from zero?– The null hypothesis is H0: ρk = 0– A suitable test statistic is:

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

Eq. 9.17 0 0,11

kk

rZ T r NT



For our problem, we have:

–We reject the hypotheses H0: ρ1 = 0 and H0: ρ2 = 0

–We have insufficient evidence to reject H0: ρ3 = 0

– ρ4 is on the borderline of being significant.–We conclude that G, the quarterly growth rate in U.S.

GDP, exhibits significant serial correlation at lags one and two

9.3Serial

Correlation

9.3.1aComputing

Autocorrelation

1 2

3 4

98 0.494 4.89, 98 0.414 4.10

98 0.154 1.52, 98 0.200 1.98

Z Z

Z Z



The correlogram, also called the sample autocorrelation function, is the sequence of autocorrelations r1, r2, r3, …– It shows the correlation between observations

that are one period apart, two periods apart, three periods apart, and so on

9.3Serial

Correlation

9.3.1bThe Correlagram



9.3Serial

Correlation

9.3.1bThe Correlagram

FIGURE 9.6 Correlogram for G



The correlogram can also be used to check whether the multiple regression assumption cov(et, es) = 0 for t ≠ s is violated

9.3Serial

Correlation

9.3.2Serially

Correlated Errors



Consider a model for a Phillips Curve:

– If we initially assume that inflationary expectations are constant over time (β1 = INFE

t) set β2= -γ, and add an error term:

9.3Serial

Correlation

9.3.2aA Phillips Curve

1γEt t t tINF INF U U Eq. 9.18

1 2β βt t tINF DU e Eq. 9.19



9.3Serial

Correlation


FIGURE 9.7 (a) Time series for Australian price inflation



9.3Serial

Correlation


FIGURE 9.7 (b) Time series for the quarterly change in the Australian unemployment rate



To determine if the errors are serially correlated, we compute the least squares residuals:

9.3Serial

Correlation


Eq. 9.20 1 2t t te INF b b DU



9.3Serial

Correlation


FIGURE 9.8 Correlogram for residuals from least-squares estimated Phillips curve



The k-th order autocorrelation for the residuals can be written as:

– The least squares equation is:

9.3Serial

Correlation


1

2

1

ˆ ˆ

ˆ

T

t t kt k

k T

tt

e er

e

Eq. 9.21

0.7776 0.5279

0.0658 0.2294INF DUse

Eq. 9.22



The values at the first five lags are:

9.3Serial

Correlation


1 2 3 4 50.549 0.456 0.433 0.420 0.339r r r r r



9.4 Other Tests for Serially Correlated

Errors



An advantage of this test is that it readily generalizes to a joint test of correlations at more than one lag

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test



If et and et-1 are correlated, then one way to model the relationship between them is to write:

–We can substitute this into a simple regression equation:

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test

1ρt t te e v Eq. 9.23

1 2 1β β ρt t t ty x e v Eq. 9.24



We have one complication: is unknown– Two ways to handle this are:

1. Delete the first observation and use a total of T observations

2. Set and use all T observations

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test

0e

0ˆ 0e



For the Phillips Curve:

– The results are almost identical– The null hypothesis H0: ρ = 0 is rejected at all

conventional significance levels–We conclude that the errors are serially

correlated

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test

i 6.219 38.67 -value 0.000

ii 6.202 38.47 -value 0.000

t F p

t F p



To derive the relevant auxiliary regression for the autocorrelation LM test, we write the test equation as:

– But since we know that , we get:

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test

1 2 1ˆβ β ρt t t ty x e v Eq. 9.25

1 2 ˆt t ty b b x e

1 2 1 2 1ˆ ˆβ β ρt t t t tb b x e x e v



Rearranging, we get:

– If H0: ρ = 0 is true, then LM = T x R2 has an approximate χ2

(1) distribution • T and R2 are the sample size and goodness-

of-fit statistic, respectively, from least squares estimation of Eq. 9.26

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test

1 1 2 2 1

1 2 1

ˆ ˆβ β ρˆγ γ ρ

t t t t

t t

e b b x e v

x e v

Eq. 9.26



Considering the two alternative ways to handle :

– These values are much larger than 3.84, which is the 5% critical value from a χ2

(1)-distribution• We reject the null hypothesis of no

autocorrelation– Alternatively, we can reject H0 by examining

the p-value for LM = 27.61, which is 0.000

9.4Other Tests for

Serially Correlated

Errors

9.4.1A Lagrange

Multiplier Test 0e

2

2

iii 1 89 0.3102 27.61

iv 90 0.3066 27.59

LM T R

LM T R



For a four-period lag, we obtain:

– Because the 5% critical value from a χ2(4)-

distribution is 9.49, these LM values lead us to conclude that the errors are serially correlated

9.4Other Tests for

Serially Correlated

Errors

9.4.1aTesting

Correlation at Longer Lags

2

2

iii 4 86 0.3882 33.4

iv 90 0.4075 36.7

LM T R

LM T R



This is used less frequently today because its critical values are not available in all software packages, and one has to examine upper and lower critical bounds instead– Also, unlike the LM and correlogram tests, its

distribution no longer holds when the equation contains a lagged dependent variable

9.4Other Tests for

Serially Correlated

Errors

9.4.2The Durbin-Watson Test



9.5 Estimation with Serially Correlated

Errors



Three estimation procedures are considered:1. Least squares estimation2. An estimation procedure that is relevant when

the errors are assumed to follow what is known as a first-order autoregressive model

3. A general estimation strategy for estimating models with serially correlated errors

9.5Estimation with

Serially Correlated

Errors

1ρt t te e v



We will encounter models with a lagged dependent variable, such as:

9.5Estimation with

Serially Correlated

Errors

1 1 0 1 1δ θ δ δt t t t ty y x x v



TSMR2A In the multiple regression model Where some of the xtk may be lagged values of y, vt is uncorrelated with all xtk and their past values.

1 2 2β β βt t K K ty x x v

ASSUMPTION FOR MODELS WITH A LAGGED DEPENDENT VARIABLE9.5

Estimation with Serially

Correlated Errors



Suppose we proceed with least squares estimation without recognizing the existence of serially correlated errors. What are the consequences?1. The least squares estimator is still a linear

unbiased estimator, but it is no longer best2. The formulas for the standard errors usually

computed for the least squares estimator are no longer correct• Confidence intervals and hypothesis tests

that use these standard errors may be misleading

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation



It is possible to compute correct standard errors for the least squares estimator: – HAC (heteroskedasticity and autocorrelation

consistent) standard errors, or Newey-West standard errors• These are analogous to the heteroskedasticity

consistent standard errors

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation



Consider the model yt = β1 + β2xt + et – The variance of b2 is:

where

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation

22

22

var var cov ,

cov ,var 1

var

t t t s t st t s

t s t st s

t tt t t

t

b w e w w e e

w w e ew e

w e

2t t tt

w x x x x

Eq. 9.27



When the errors are not correlated, cov(et, es) = 0, and the term in square brackets is equal to one. – The resulting expression

is the one used to find heteroskedasticity-consistent (HC) standard errors

–When the errors are correlated, the term in square brackets is estimated to obtain HAC standard errors

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation

22var vart tt

b w e



If we call the quantity in square brackets g and its estimate , then the relationship between the two estimated variances is:

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation

2 2 ˆvar varHAC HCb b g

g

Eq. 9.28



Let’s reconsider the Phillips Curve model:

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation

0.7776 0.5279 0.0658 0.2294 incorrect se

0.1030 0.3127 HAC se

INF DU

Eq. 9.29



The t and p-values for testing H0: β2 = 0 are:

9.5Estimation with

Serially Correlated

Errors

9.5.1Least Squares

Estimation

0.5279 0.2294 2.301 0.0238 from LS standard errors

0.5279 0.3127 1.688 0.0950 from HAC standard errors

t p

t p



Return to the Lagrange multiplier test for serially correlated errors where we used the equation:

– Assume the vt are uncorrelated random errors with zero mean and constant variances:

9.5Estimation with

Serially Correlated

Errors

9.5.2Estimating an

AR(1) Error Model

1ρt t te e v Eq. 9.30

20 var cov , 0 for t t v t sE v v v v t s Eq. 9.31



Eq. 9.30 describes a first-order autoregressive model or a first-order autoregressive process for et

– The term AR(1) model is used as an abbreviation for first-order autoregressive model

– It is called an autoregressive model because it can be viewed as a regression model

– It is called first-order because the right-hand-side variable is et lagged one period

9.5Estimation with

Serially Correlated

Errors

9.5.2Estimating an

AR(1) Error Model



We assume that:

The mean and variance of et are:

The covariance term is:

9.5Estimation with

Serially Correlated

Errors

9.5.2aProperties of an

AR(1) Error

1 ρ 1 Eq. 9.32

2

220 var

1 ρv

t t eE e e

Eq. 9.33

2

2

ρcov , , 01 ρ

kv

t t ke e k

Eq. 9.34



The correlation implied by the covariance is:

9.5Estimation with

Serially Correlated

Errors

2 2

2 2

ρ corr ,

cov ,

var var

cov ,var

ρ 1 ρ

1 ρ

ρ

k t t k

t t k

t t k

t t k

t

kv

v

k

e e

e e

e e

e ee

Eq. 9.35


AR(1) Error



Setting k = 1:

– ρ represents the correlation between two errors that are one period apart• It is the first-order autocorrelation for e,

sometimes simply called the autocorrelation coefficient• It is the population autocorrelation at lag one for a

time series that can be described by an AR(1) model• r1 is an estimate for ρ when we assume a series is

AR(1)

9.5Estimation with

Serially Correlated

Errors

1 1ρ corr , ρt te e Eq. 9.36


AR(1) Error



Each et depends on all past values of the errors vt:

– For the Phillips Curve, we find for the first five lags:

– For an AR(1) model, we have:

9.5Estimation with

Serially Correlated

Errors

2 31 2 3ρ ρ ρt t t t te v v v v Eq. 9.37

1 2 3 4 50.549 0.456 0.433 0.420 0.339r r r r r

1 1ˆ ˆρ ρ 0.549r


AR(1) Error



For longer lags, we have:

9.5Estimation with

Serially Correlated

Errors

222

333

444

555

ˆ ˆρ ρ 0.549 0.301

ˆ ˆρ ρ 0.549 0.165

ˆ ˆρ ρ 0.549 0.091

ˆ ˆρ ρ 0.549 0.050


AR(1) Error



Our model with an AR(1) error is:

with -1 < ρ < 1– For the vt, we have:

9.5Estimation with

Serially Correlated

Errors

9.5.2bNonlinear Least

Squares Estimation

1 2 1β β with ρt t t t t ty x e e e v Eq. 9.38

210 var cov , 0 for t t v t tE v v v v t s Eq. 9.39



With the appropriate substitutions, we get:

– For the previous period, the error is:

–Multiplying by ρ:

9.5Estimation with

Serially Correlated

Errors


Squares Estimation

1 2 1β β ρt t t ty x e v Eq. 9.40

1 1 1 2 1β βt t te y x Eq. 9.41

1 1 1 2 1ρ ρβ ρβt t t te e y x Eq. 9.42



Substituting, we get:

9.5Estimation with

Serially Correlated

Errors


Squares Estimation

1 2 1 2 1β 1 ρ β ρ ρβt t t t ty x y x v Eq. 9.43



The coefficient of xt-1 equals -ρβ2 – Although Eq. 9.43 is a linear function of the

variables xt , yt-1 and xt-1, it is not a linear function of the parameters (β1, β2, ρ)

– The usual linear least squares formulas cannot be obtained by using calculus to find the values of (β1, β2, ρ) that minimize Sv

• These are nonlinear least squares estimates

9.5Estimation with

Serially Correlated

Errors


Squares Estimation



Our Phillips Curve model assuming AR(1) errors is:

– Applying nonlinear least squares and presenting the estimates in terms of the original untransformed model, we have:

9.5Estimation with

Serially Correlated

Errors


Squares Estimation

1 2 1 2 1β 1 ρ β ρ ρβt t t t tINF DU INF DU v Eq. 9.44

10.7609 0.6944 0.557

0.1245 0.2479 0.090t t tINF DU e e v

se

Eq. 9.45



Nonlinear least squares estimation of Eq. 9.43 is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure

9.5Estimation with

Serially Correlated

Errors

9.5.2cGeneralized

Least Squares Estimation



We have the model:

– Suppose now that we consider the model:

• This new notation will be convenient when we discuss a general class of autoregressive distributed lag (ARDL) models–Eq. 9.47 is a member of this class

9.5Estimation with

Serially Correlated

Errors

9.5.3Estimating a More General

Model

1 2 1 2 1β 1 ρ β ρ ρβt t t t ty x y x v Eq. 9.46

1 1 0 1 1δ θ δ δt t t t ty y x x v Eq. 9.47



Note that Eq. 9.47 is the same as Eq. 9.47 since:

– Eq. 9.46 is a restricted version of Eq. 9.47 with the restriction δ1 = -θ1δ0 imposed

9.5Estimation with

Serially Correlated

Errors


Model

1 0 2 1 2 1δ β 1 ρ δ β δ ρβ θ ρ Eq. 9.48



Applying the least squares estimator to Eq. 9.47 using the data for the Phillips curve example yields:

9.5Estimation with

Serially Correlated

Errors


Model

1 10.3336 0.5593 0.6882 0.3200

0.0899 0.0908 0.2575 0.2499t t t tINF INF DU DU

se

Eq. 9.49



The equivalent AR(1) estimates are:

– These are similar to our other estimates

9.5Estimation with

Serially Correlated

Errors


Model

1

1

0 2

1 2

ˆ ˆ ˆδ β 1 ρ 0.7609 1 0.5574 0.3368ˆ ˆθ ρ 0.5574ˆ ˆδ β 0.6944ˆ ˆˆδ ρβ 0.5574 0.6944 0.3871



The original economic model for the Phillips Curve was:

– Re-estimation of the model after omitting DUt-1 yields:

9.5Estimation with

Serially Correlated

Errors


Model

1γEt t t tINF INF U U Eq. 9.50

10.3548 0.5282 0.4909

0.0876 0.0851 0.1921t t tINF INF DU

se

Eq. 9.51



In this model inflationary expectations are given by:

– A 1% rise in the unemployment rate leads to an approximate 0.5% fall in the inflation rate

9.5Estimation with

Serially Correlated

Errors


Model

10.3548 0.5282Et tINF INF



We have described three ways of overcoming the effect of serially correlated errors:1. Estimate the model using least squares with

HAC standard errors2. Use nonlinear least squares to estimate the

model with a lagged x, a lagged y, and the restriction implied by an AR(1) error specification

3. Use least squares to estimate the model with a lagged x and a lagged y, but without the restriction implied by an AR(1) error specification

9.5Estimation with

Serially Correlated

Errors

9.5.4Summary of

Section 9.5 and Looking Ahead



9.6 Autoregressive Distributed Lag

Models



An autoregressive distributed lag (ARDL) model is one that contains both lagged xt’s and lagged yt’s

– Two examples:

9.6Autoregressive Distributed Lag

Models

0 1 1 1 1t t t q t q t p t p ty x x x y y v Eq. 9.52

1 1

1

ADRL 1,1 : 0.3336 0.5593 0.6882 0.3200

ADRL 1,0 : 0.3548 0.5282 0.4909

t t t t

t t t

INF INF DU DU

INF INF DU



An ARDL model can be transformed into one with only lagged x’s which go back into the infinite past:

– This model is called an infinite distributed lag model


Models

0 1 1 2 2 3 3

0

β β β

β

t t t t t t

s t s ts

y x x x x e

x e

Eq. 9.53



Four possible criteria for choosing p and q:1. Has serial correlation in the errors been

eliminated?2. Are the signs and magnitudes of the estimates

consistent with our expectations from economic theory?

3. Are the estimates significantly different from zero, particularly those at the longest lags?

4. What values for p and q minimize information criteria such as the AIC and SC?


Models



The Akaike information criterion (AIC) is:

where K = p + q + 2The Schwarz criterion (SC), also known as the Bayes information criterion (BIC), is:

– Because Kln(T)/T > 2K/T for T ≥ 8, the SC penalizes additional lags more heavily than does the AIC


Models

2AIC ln SSE KT T

Eq. 9.54

lnSC ln

K TSSET T

Eq. 9.55



Consider the previously estimated ARDL(1,0) model:


Models

Eq. 9.56

9.6.1The Phillips

Curve

10.3548 0.5282 0.4909 , obs 90

0.0876 0.0851 0.1921t t tINF INF DU

se




Models

9.6.1The Phillips

Curve

FIGURE 9.9 Correlogram for residuals from Phillips curve ARDL(1,0) model




Models

9.6.1The Phillips

Curve

Table 9.3 p-values for LM Test for Autocorrelation



For an ARDL(4,0) version of the model:


Models

Eq. 9.57

9.6.1The Phillips

Curve

1 2 3

-4

0.1001 0.2354 0.1213 0.1677

0.0983 0.1016 0.1038 0.1050

0.2819 0.7902

0.1014 0.1885

t t t t

t t

INF INF INF INF

se

INF DU

obs 87



Inflationary expectations are given by:


Models

9.6.1The Phillips

Curve

1 2 3 -40.1001 0.2354 0.1213 0.1677 0.2819Et t t t tINF INF INF INF INF




Models

9.6.1The Phillips

Curve

Table 9.4 AIC and SC Values for Phillips Curve ARDL Models



Recall the model for Okun’s Law:


Models

9.6.2Okun’s Law

Eq. 9.58

1 20.5836 0.2020 0.1653 0.0700G , obs 96

0.0472 0.0324 0.0335 0.0331t t t tDU G G

se




ModelsFIGURE 9.10 Correlogram for residuals from Okun’s law ARDL(0,2) model

9.6.2Okun’s Law




ModelsTable 9.5 AIC and SC Values for Okun’s Law ARDL Models

9.6.2Okun’s Law



Now consider this version:


Models

9.6.2Okun’s Law

Eq. 9.59

1 10.3780 0.3501 0.1841 0.0992G , obs 96

0.0578 0.0846 0.0307 0.0368t t t tDU DU G

se



An autoregressive model of order p, denoted AR(p), is given by:


Models

9.6.3Autoregressive

Models

Eq. 9.60 1 1 2 2δ θ θ θt t t p t p ty y y y v



Consider a model for growth in real GDP:


Models

9.6.3Autoregressive

Models

Eq. 9.61

1 20.4657 0.3770 0.2462

0.1433 0.1000 0.1029 obs = 96t t tG G G

se




Models

9.6.3Autoregressive

Models

FIGURE 9.11 Correlogram for residuals from AR(2) model for GDP growth




Models

9.6.3Autoregressive

Models

Table 9.6 AIC and SC Values for AR Model of Growth in U.S. GDP



9.7 Forecasting



We consider forecasting using three different models:1. AR model2. ARDL model3. Exponential smoothing model

9.7Forecasting



Consider an AR(2) model for real GDP growth:

The model to forecast GT+1 is:

9.7Forecasting

9.7.1Forecasting with

an AR Model

Eq. 9.62 1 1 2 2δ θ θt t t tG G G v

1 1 2 1 1δ θ θT T T TG G G v



The growth values for the two most recent quarters are:

GT = G2009Q3 = 0.8GT-1 = G2009Q2 = -0.2

The forecast for G2009Q4 is:

9.7Forecasting


an AR Model

Eq. 9.63 1 1 2 1

ˆˆ ˆ ˆδ θ θ

0.46573 0.37700 0.8 0.24624 0.2

0.7181

T T TG G G



For two quarters ahead, the forecast for G2010Q1 is:

For three periods out, it is:

9.7Forecasting


an AR Model

Eq. 9.642 1 1 2

ˆˆ ˆ ˆδ θ θ0.46573 0.37700 0.71808 0.24624 0.80.9334

T T TG G G

Eq. 9.653 1 2 2 1

ˆˆ ˆ ˆδ θ θ0.46573 0.37700 0.93343 0.24624 0.718080.9945

T T TG G G



Summarizing our forecasts:– Real GDP growth rates for 2009Q4, 2010Q1,

and 2010Q2 are approximately 0.72%, 0.93%, and 0.99%, respectively

9.7Forecasting


an AR Model



A 95% interval forecast for j periods into the future is given by:

where is the standard error of the forecast error and df is the number of degrees of freedom in the estimation of the AR model

9.7Forecasting


an AR Model

0.975,ˆ σT j jdfG t

σ j



The first forecast error, occurring at time T+1, is:

Ignoring the error from estimating the coefficients, we get:

9.7Forecasting


an AR Model

1 1 1 1 1 2 2 1 1ˆˆ ˆ ˆδ δ θ θ θ θT T T T Tu G G G G v

1 1Tu v Eq. 9.66



The forecast error for two periods ahead is:

The forecast error for three periods ahead is:

9.7Forecasting


an AR Model

2 1 1 1 2 1 1 2 1 1 2ˆθ θ θT T T T T Tu G G v u v v v Eq. 9.67

23 1 2 2 1 3 1 2 1 1 2 3θ θ θ θ θT T T Tu u u v v v v Eq. 9.68



Because the vt’s are uncorrelated with constant variance , we can show that:

9.7Forecasting


an AR Model

2 21 1

2 2 22 2 1

22 2 2 23 3 1 2 1

σ var σ

σ var σ 1 θ

σ var σ θ θ θ 1

v

v

v

u

u

u

2v



9.7Forecasting


an AR Model

Table 9.7 Forecasts and Forecast Intervals for GDP Growth



Consider forecasting future unemployment using the Okun’s Law ARDL(1,1):

The value of DU in the first post-sample quarter is:

– But we need a value for GT+1

9.7Forecasting

9.7.2Forecasting with an ARDL Model

1 1 0 1 1δ θ δ δt t t t tDU DU G G v Eq. 9.69

1 1 0 1 1 1δ θ δ δT T T T TDU DU G G v Eq. 9.70



Now consider the change in unemployment– Rewrite Eq. 9.70 as:

– Rearranging:

9.7Forecasting


1 1 1 0 1 1 1δ θ δ δT T T T T T TU U U U G G v

Eq. 9.71 1 1 1 1 0 1 1 1

* *1 2 1 0 1 1 1

δ θ 1 θ δ δ

δ θ θ δ δT T T T T T

T T T T T

U U U G G v

U U G G v



For the purpose of computing point and interval forecasts, the ARDL(1,1) model for a change in unemployment can be written as an ARDL(2,1) model for the level of unemployment– This result holds not only for ARDL models

where a dependent variable is measured in terms of a change or difference, but also for pure AR models involving such variables

9.7Forecasting




Another popular model used for predicting the future value of a variable on the basis of its history is the exponential smoothing method– Like forecasting with an AR model, forecasting

using exponential smoothing does not utilize information from any other variable

9.7Forecasting

9.7.3Exponential Smoothing



One possible forecasting method is to use the average of past information, such as:

– This forecasting rule is an example of a simple (equally-weighted) moving average model with k = 3

9.7Forecasting


1 21ˆ

3T T T

Ty y yy



Now consider a form in which the weights decline exponentially as the observations get older:

–We assume that 0 < α < 1– Also, it can be shown that:

9.7Forecasting


1 21 T 1 2ˆ αy α 1 α α 1 αT T Ty y y Eq. 9.72

0α 1 α 1

s

s



For forecasting, recognize that:

–We can simplify to:

9.7Forecasting


2 31 2 3ˆ1 α α 1 α α 1 α α 1 αT T T Ty y y y Eq. 9.73

1ˆ ˆα 1 αT T Ty y y Eq. 9.74



The value of α can reflect one’s judgment about the relative weight of current information– It can be estimated from historical information

by obtaining within-sample forecasts:

• Choosing α that minimizes the sum of squares of the one-step forecast errors:

9.7Forecasting


1 1ˆ ˆα 1 α 2,3, ,t t ty y y t T Eq. 9.75

1 1ˆ ˆα + 1 αt t t t t tv y y y y y Eq. 9.76



9.7Forecasting


FIGURE 9.12 (a) Exponentially smoothed forecasts for GDP growth with α = 0.38



9.7Forecasting


FIGURE 9.12 (b) Exponentially smoothed forecasts for GDP growth with α = 0.8



The forecasts for 2009Q4 from each value of α are:

9.7Forecasting


1

1

ˆ ˆα 0.38 : α 1 α 0.38 0.8 1 0.38 0.403921

= 0.0536ˆ ˆα 0.8 : α 1 α 0.8 0.8 1 0.8 0.393578

= 0.5613

T T T

T T T

G G G

G G G



9.8 Multiplier Analysis



Multiplier analysis refers to the effect, and the timing of the effect, of a change in one variable on the outcome of another variable

9.8Multiplier Analysis



Let’s find multipliers for an ARDL model of the form:

–We can transform this into an infinite distributed lag model:


1 1 0 1 1t t p t p t t q t q ty y y x x x v Eq. 9.77

0 t 1 1 2 2 3 3α β x + β β βt t t t ty x x x e Eq. 9.78



The multipliers are defined as:


0

0

β period delay multiplier

β period interim multiplier

β total multiplier

ts

t s

s

jj

jj

ys

x

s



The lag operator is defined as:

– Lagging twice, we have:

– Or:

–More generally, we have:


1t tLy y

1 2t t tL Ly Ly y

22t tL y y

st t sL y y



Now rewrite our model as:


2 21 2 0 1 2

pt t t p t t t t

qq t t

y Ly L y L y x Lx L x

L x v

Eq. 9.79



Rearranging terms:


2 21 2 0 1 21 p q

p t q t tL L L y L L L x v Eq. 9.80



Let’s apply this to our Okun’s Law model– The model:

can be rewritten as:


Eq. 9.81 1 1 0 1 1δ θ δ δt t t t tDU DU G G v

Eq. 9.82 1 0 11 θ δ δ δt t tL DU L G v



Define the inverse of (1 – θ1L) as (1 – θ1L)-1 such that:


11 11 θ 1 θ 1L L



Multiply both sides of Eq. 9.82 by (1 – θ1L)-1:

– Equating this with the infinite distributed lag representation:


1 1 11 1 0 1 11 θ δ 1 θ δ δ 1 θt t tDU L L L G L v Eq. 9.83

Eq. 9.84 0 1 1 2 2 3 3

2 30 1 2 3

α β β β β

α β β β βt t t t t t

t t

DU G G G G e

L L L G e



For Eqs. 9.83 and 9.84 to be identical, it must be true that:


Eq. 9.85

Eq. 9.86

11

12 30 1 2 3 1 0 1

11

α= 1 θ δ

β β β β 1 θ δ δ

1 θt t

L

L L L L L

e L v

Eq. 9.87



Multiply both sides of Eq. 9.85 by (1 – θ1L) to obtain (1 – θ1L)α = δ. – Note that the lag of a constant that does not

change so Lα = α– Now we have:


11

δ1 θ α δ and α1 θ



Multiply both sides of Eq. 9.86 by (1 – θ1L):


2 30 1 1 0 1 2 3

2 30 1 2 3

2 30 1 1 1 2 1

2 30 1 0 1 2 1 1 3 2 1

δ δ 1 θ β β β β

β β β β

β θ β θ β θ

β β β θ β β θ β β θ

L L L L L

L L L

L L L

L L L

Eq. 9.88



Rewrite Eq. 9.86 as:

– Equating coefficients of like powers in L yields:

and so on


2 3 2 30 1 0 1 0 1 2 1 1 3 2 1δ δ 0 0 β β β θ β β θ β β θL L L L L L Eq. 9.89

0 0

1 1 0 1

2 1 1

3 2 1

δ = βδ β β θ0 β β θ0 β β θ



We can now find the β’s using the recursive equations:


Eq. 9.90

0 0

1 1 0 1

1 1

β = δβ δ β θβ β θ for 2j j j



You can start from the equivalent of Eq. 9.88 which, in its general form, is:

– Given the values p and q for your ARDL model, you need to multiply out the above expression, and then equate coefficients of like powers in the lag operator


Eq. 9.91

2 20 1 2 1 2

2 30 1 2 3

δ δ δ δ 1 θ θ θ

β β β β

q pq pL L L L L L

L L L



For the Okun’s Law model:

– The impact and delay multipliers for the first four quarters are:


1 10.3780 0.3501 0.1841 0.0992t t t tDU DU G G

0 0

1 1 0 1

2 1 1

3 2 1

4 3 1

ˆ ˆβ = δ 0.1841ˆ ˆ ˆ ˆβ δ β θ 0.099155 0.184084 0.350116 0.1636ˆ ˆ ˆβ β θ 0.163606 0.350166 0.0573ˆ ˆ ˆβ β θ 0.057281 0.350166 0.0201ˆ ˆ ˆβ β θ 0.020055 0.350166 0.0070




FIGURE 9.13 Delay multipliers from Okun’s law ARDL(1,1) model



We can estimate the total multiplier given by:

and the normal growth rate that is needed to maintain a constant rate of unemployment:


0

β jj

0

α βN jj

G



We can show that:

– An estimate for α is given by:

– Therefore, normal growth rate is:


1 0 10

0 1

ˆ ˆ ˆδ δ θ 0.163606ˆβ δ 0.184084 0.4358ˆ 1 0.3501161 θjj

1

δ 0.37801α 0.5817ˆ 0.6498841 θ

N0.5817G 1.3% per quarter0.4358



Key Words



AIC criterionAR(1) errorAR(p) modelARDL(p,q) modelautocorrelationAutoregressive distributed lagsautoregressive errorautoregressive modelBIC criterioncorrelogramdelay multiplierdistributed lag weight

Keywords

dynamic modelsexponential smoothingfinite distributed lagforecast errorforecast intervalsforecastingHAC standard errorsimpact multiplierinfinite distributed laginterim multiplierlag lengthlag operatorlagged dependent variable

LM testmultiplier analysisnonlinear least squaresout-of-sample forecastssample autocorrelationsserial correlation standard error of forecast errorSC criteriontotal multiplierT x R2 form of LM test within-sample forecasts



Appendices



For the Durbin-Watson test, the hypotheses are:

The test statistic is:

9AThe Durbin-Watson Test

Eq. 9A.1

0 1: 0 : 0H H

21

2

2

1

ˆ ˆ

ˆ

T

t tt

T

tt

e ed

e



We can expand the test statistic as:


Eq. 9A.2

2 21 1

2 2 2

2

1

2 21 1

2 2 2

2 2 2

1 1 1

1

ˆ ˆ ˆ ˆ2

ˆ

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ

1 1 2

T T T

t t t tt t t

T

tt

T T T

t t t tt t tT T T

t t tt t t

e e e ed

e

e e e e

e e e

r



We can now write:

– If the estimated value of ρ is r1 = 0, then the Durbin-Watson statistic d ≈ 2• This is taken as an indication that the model

errors are not autocorrelated– If the estimate of ρ happened to be r1 = 1 then d ≈ 0• A low value for the Durbin-Watson statistic

implies that the model errors are correlated, and ρ > 0


Eq. 9A.3 12 1d r



9AThe Durbin-Watson Test FIGURE 9A.1 Testing for positive autocorrelation




9A.1The Durbin-

Watson Bounds Test

FIGURE 9A.2 Upper and lower critical value bounds for the Durbin-Watson test



Decision rules, known collectively as the Durbin-Watson bounds test:– If d < dLc: reject H0: ρ = 0 and accept H1:

ρ > 0– If d > dUc do not reject H0: ρ = 0 – If dLc < d < dUc, the test is inconclusive


9A.1The Durbin-

Watson Bounds Test



Note that:

Further substitution shows that:

9BProperties of

the AR(1) Error

1

2 1

22 1

ρ

ρ ρ

ρ ρ

t t t

t t t

t t t

e e v

e v v

e v v

Eq. 9B.1

23 2 1

3 23 2 1

ρ ρ ρ

ρ ρ ρt t t t t

t t t t

e e v v v

e v v v

Eq. 9B.2



Repeating the substitution k times and rearranging:

If we let k → ∞, then we have:

9BProperties of

the AR(1) Error

2 11 2 1ρ ρ ρ ρk k

t t k t t t t ke e v v v v Eq. 9B.3

2 31 2 3ρ ρ ρt t t t te v v v v Eq. 9B.4



We can now find the properties of et:

9BProperties of

the AR(1) Error

2 31 2 3

2 3

ρE ρ ρ

0 ρ 0 ρ 0 ρ 00

t t t t tE e E v v E v E v

2 4 61 2 3

2 2 2 4 2 6 2

2 2 4 6

2

2

var var ρ var ρ var ρ var

ρ ρ ρ

1 ρ ρ ρ

1 ρ

t t t t t

v v v v

v

v

e v v v v



The covariance for one period apart is:

9BProperties of

the AR(1) Error

1

2 31 2 3

2 31 2 3 4

2 3 2 5 21 2 3

2 2 4

2

2

cov ,

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ

ρ 1 ρ ρ

ρ1 ρ

t t t t t

t t t t

t t t t

t t t

v

v

e e E e e

E v v v v

v v v v

E v E v E v



Similarly, the covariance for k periods apart is:

9BProperties of

the AR(1) Error

2

2

ρcov , 0

1 ρ

kv

t t ke e k



We are considering the simple regression model with AR(1) errors:

To specify the transformed model we begin with:

– Rearranging terms:

9CGeneralized


1 2 1 t t t t t ty x e e e v

1 2 1 1 2 1t t t t ty x y x v Eq. 9C.1

1 1 2 11t t t t ty y x x v Eq. 9C.2



Defining the following transformed variables:

Substituting the transformed variables, we get:

9CGeneralized


Eq. 9C.3

1 2 1 1 1t t t t t t ty y y x x x x

1 1 2 2t t t ty x x v



There are two problems:1. Because lagged values of yt and xt had to be

formed, only (T - 1) new observations were created by the transformation

2. The value of the autoregressive parameter ρ is not known

9CGeneralized




For the second problem, we can write Eq. 9C.1 as:

For the first problem, note that:

and that

9CGeneralized


Eq. 9C.4 1 2 1 1 2 1( )t t t t ty x y x v

1 1 1 2 1y x e

2 2 2 21 1 1 2 11 1 1 1y x e



Or:

where

9CGeneralized


1 11 1 12 2 1y x x e Eq. 9C.5

2 21 1 11

2 212 1 1 1

1 1

1 1

y y x

x x e e

Eq. 9C.6



To confirm that the variance of e*1 is the same as

that of the errors (v2, v3,…, vT), note that:

9CGeneralized


22 2 2

1 1 2var( ) (1 ) var( ) (1 )1

vve e

chapter 9 regression with time series data: stationary variables

Documents

time series data

timeseries observations

correlatedtimeseries

stationary variables9

stationary variablesways

number of time periods

future time periods9

regression of yt