Using SAS for Time Series Data LSU Economics Department March 16, 2012.

Using SAS for Time Series Data

LSU Economics DepartmentMarch 16, 2012

Next Workshop March 30

Instrumental Variables Estimation

Principles of Econometrics, 4th Edition

Chapter 12: Regression with Time-Series Data:

Nonstationary Variables

Time-Series Data:Nonstationary Variables




12.1 Stationary and Nonstationary Variables12.2 Spurious Regressions12.3 Unit Root Tests for Nonstationarity

Chapter Contents




The aim is to describe how to estimate regression models involving nonstationary variables– The first step is to examine the time-series

concepts of stationarity (and nonstationarity) and how we distinguish between them.




12.1

Stationary and Nonstationary Variables




The change in a variable is an important concept

– The change in a variable yt, also known as its first difference, is given by Δyt = yt – yt-1.

• Δyt is the change in the value of the variable y from period t - 1 to period t

12.1Stationary and Nonstationary

Variables





VariablesFIGURE 12.1 U.S. economic time series





VariablesFIGURE 12.1 (Continued) U.S. economic time series




Formally, a time series yt is stationary if its mean and variance are constant over time, and if the covariance between two values from the series depends only on the length of time separating the two values, and not on the actual times at which the variables are observed


Variables




That is, the time series yt is stationary if for all values, and every time period, it is true that:


Variables

2

μ (constant mean)

var σ (constant variance)

cov , cov , γ (covariance depends on , not )

t

t

t t s t t s s

E y

y

y y y y s t

Eq. 12.1a

Eq. 12.1b

Eq. 12.1c





Variables

12.1.1The First-Order Autoregressive

Model

FIGURE 12.2 Time-series models





Variables

12.1.1The First-Order Autoregressive

Model

FIGURE 12.2 (Continued) Time-series models




12.2Spurious

Regressions FIGURE 12.3 Time series and scatter plot of two random walk variables




A simple regression of series one (rw1) on series two (rw2) yields:

– These results are completely meaningless, or spurious • The apparent significance of the relationship

is false

12.2Spurious

Regressions

21 217.818 0.842 , 0.70

( ) (40.837)t trw rw R

t




When nonstationary time series are used in a regression model, the results may spuriously indicate a significant relationship when there is none– In these cases the least squares estimator and

least squares predictor do not have their usual properties, and t-statistics are not reliable

– Since many macroeconomic time series are nonstationary, it is particularly important to take care when estimating regressions with macroeconomic variables

12.2Spurious

Regressions




12.3

Unit Root Tests for Stationarity




There are many tests for determining whether a series is stationary or nonstationary– The most popular is the Dickey–Fuller test

12.3Unit Root Tests for

Stationarity




Consider the AR(1) model:

–We can test for nonstationarity by testing the null hypothesis that ρ = 1 against the alternative that |ρ| < 1• Or simply ρ < 1


Stationarity

1t t ty y v Eq. 12.4

12.3.1Dickey-Fuller Test 1

(No constant and No Trend)




A more convenient form is:

– The hypotheses are:


Stationarity

Eq. 12.5a 1 1 1

1

1

1t t t t t

t t t

t t

y y y y v

y y v

y v

0 0

1 1

: 1 : 0

: 1 : 0

H H

H H

12.3.1Dickey-Fuller Test 1

(No constant and No Trend)




The second Dickey–Fuller test includes a constant term in the test equation:

– The null and alternative hypotheses are the same as before


Stationarity

12.3.2Dickey-Fuller Test 2 (With Constant but

No Trend)

Eq. 12.5b 1t t ty y v




The third Dickey–Fuller test includes a constant and a trend in the test equation:

– The null and alternative hypotheses are H0: γ = 0 and H1:γ < 0


Stationarity

12.3.3Dickey-Fuller Test 3 (With Constant and

With Trend)

Eq. 12.5c 1t t ty y t v




To test the hypothesis in all three cases, we simply estimate the test equation by least squares and examine the t-statistic for the hypothesis that γ = 0 – Unfortunately this t-statistic no longer has the

t-distribution– Instead, we use the statistic often called a τ

(tau) statistic


Stationarity

12.3.4The Dickey-Fuller

Critical Values





Stationarity


Critical Values

Table 12.2 Critical Values for the Dickey–Fuller Test




To carry out a one-tail test of significance, if τc is the critical value obtained from Table 12.2, we reject the null hypothesis of nonstationarity if τ ≤ τc

– If τ > τc then we do not reject the null hypothesis that the series is nonstationary


Stationarity


Critical Values




An important extension of the Dickey–Fuller test allows for the possibility that the error term is autocorrelated– Consider the model:

where


Stationarity


Critical Values

11

m

t t s t s ts

y y a y v

Eq. 12.6

1 1 2 2 2 3, ,t t t t t ty y y y y y




As an example, consider the two interest rate series:

– The federal funds rate (Ft)

– The three-year bond rate (Bt)

Following procedures described in Sections 9.3 and 9.4, we find that the inclusion of one lagged difference term is sufficient to eliminate autocorrelation in the residuals in both cases


Stationarity

12.3.6The Dickey-Fuller Tests: An Example




The results from estimating the resulting equations are:

– The 5% critical value for tau (τc) is -2.86

– Since -2.505 > -2.86, we do not reject the null hypothesis


Stationarity

12.3.6The Dickey-Fuller Tests: An Example

1 1

1 1

0.173 0.045 0.561

( ) ( 2.505)

0.237 0.056 0.237

( ) ( 2.703)

t t t

t t t

F F F

tau

B B B

tau




Recall that if yt follows a random walk, then γ = 0 and the first difference of yt becomes:

– Series like yt, which can be made stationary by taking the first difference, are said to be integrated of order one, and denoted as I(1)• Stationary series are said to be integrated of

order zero, I(0)– In general, the order of integration of a series is

the minimum number of times it must be differenced to make it stationary


Stationarity

12.3.7Order of Integration

1t t t ty y y v




The results of the Dickey–Fuller test for a random walk applied to the first differences are:


Stationarity


10.447

( ) ( 5.487)

t tF F

tau

10.701

( ) ( 7.662)

t tB B

tau




Based on the large negative value of the tau statistic (-5.487 < -1.94), we reject the null hypothesis that ΔFt is nonstationary and accept the alternative that it is stationary

–We similarly conclude that ΔBt is stationary (-7:662 < -1:94)


Stationarity





12.4

Cointegration




As a general rule, nonstationary time-series variables should not be used in regression models to avoid the problem of spurious regression– There is an exception to this rule

12.4Cointegration




There is an important case when et = yt - β1 - β2xt is a stationary I(0) process

– In this case yt and xt are said to be cointegrated

• Cointegration implies that yt and xt share similar stochastic trends, and, since the difference et is stationary, they never diverge too far from each other

12.4Cointegration




The test for stationarity of the residuals is based on the test equation:

– The regression has no constant term because the mean of the regression residuals is zero.

–We are basing this test upon estimated values of the residuals

12.4Cointegration

1ˆ ˆγt t te e v Eq. 12.7




12.4Cointegration Table 12.4 Critical Values for the Cointegration Test




There are three sets of critical values–Which set we use depends on whether the

residuals are derived from:

12.4Cointegration

Eq. 12.8a ˆ1: t t tEquation e y bx

2 1ˆ2 : t t tEquation e y b x b

2 1ˆˆ3: t t tEquation e y b x b t

Eq. 12.8b

Eq. 12.8c




Consider the estimated model:

– The unit root test for stationarity in the estimated residuals is:

12.4Cointegration

Eq. 12.9

12.4.1An Example of a

Cointegration Test

2ˆ 1.140 0.914 , 0.881

( ) (6.548) (29.421)t tB F R

t

1 1ˆ ˆ ˆ0.225 0.254

( ) ( 4.196)t t te e e

tau




The null and alternative hypotheses in the test for cointegration are:

– Similar to the one-tail unit root tests, we reject the null hypothesis of no cointegration if τ ≤ τc, and we do not reject the null hypothesis that the series are not cointegrated if τ > τc

12.4Cointegration

12.4.1An Example of a

Cointegration Test

0

1

: the series are not cointegrated residuals are nonstationary

: the series are cointegrated residuals are stationary

H

H




Chapter 9Regression with Time Series

Data:Stationary Variables

Walter R. Paczkowski Rutgers University


Chapter 9: Regression with Time Series Data: Stationary Variables

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. 9.21( , )t t ty f y x

Eq. 9.31 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




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 y

x x

Eq. 9.7

9.2Finite

Distributed Lags




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



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



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 00,1

1k

k

rZ T r N

T



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



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

9.3Serial

Correlation

9.3.2aA Phillips Curve

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



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 0.2294

INF DU

se

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



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



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 ρ

ˆγ γ ρt t t t

t t

e b b x e v

x e v

Eq. 9.26




9.5

Estimation with Serially Correlated Errors



Three estimation procedures are considered:

1. Least squares estimation

2. 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 VARIABLE

9.5Estimation 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 best

2. 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

2

t t ttw 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



Substituting, we get:

9.5Estimation with

Serially Correlated

Errors

9.5.2bNonlinear Least

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

9.5.2bNonlinear Least

Squares Estimation

Using SAS for Time Series Data LSU Economics Department March 16, 2012.

Documents

Transcript of Using SAS for Time Series Data LSU Economics Department March 16, 2012.