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Chapter 9 Regression with Time Series Data: Stationary Variables Walter R. Paczkowski Rutgers University Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 1 Chapter Contents 9.1 Introduction 9.2 Finite Distributed Lags 9.3 Serial Correlation 9.4 Other Tests for Serially Correlated Errors 9.5 Estimation with Serially Correlated Errors 9.6 Autoregressive Distributed Lag Models 9.7 Forecasting 9.8 Multiplier Analysis Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 2 9.1 Introduction Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 3 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 4 9.1 Introduction 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 5 9.1 Introduction Principles of Econometrics, 4th Edition FIGURE 9.1 The distributed lag effect Chapter 9: Regression with Time Series Data: Stationary Variables Page 6 9.1 Introduction 9.1.1 Dynamic Nature of Relationships 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 yt f ( xt , xt 1 , xt 2 ,...) Eq. 9.1 • Because of the existence of these lagged effects, Eq. 9.1 is called a distributed lag model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 7 9.1 Introduction 9.1.1 Dynamic Nature of Relationships Eq. 9.2 Ways to model the dynamic relationship (Continued): 2. Capturing the dynamic characteristics of timeseries by specifying a model with a lagged dependent variable as one of the explanatory variables yt f ( yt 1 , xt ) • Or have: Eq. 9.3 yt f ( yt 1 , xt , xt 1 , xt 2 ) – Such models are called autoregressive distributed lag (ARDL) models, with ‘‘autoregressive’’ meaning a regression of yt on its own lag or lags Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 8 9.1 Introduction 9.1.1 Dynamic Nature of Relationships Ways to model the dynamic relationship (Continued): 3. Model the continuing impact of change over several periods via the error term yt f ( xt ) et Eq. 9.4 et f (et 1 ) • In this case et is correlated with et - 1 • We say the errors are serially correlated or autocorrelated Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 9 9.1 Introduction 9.1.2 Least Squares Assumptions The primary assumption is Assumption MR4: cov yi , y j cov ei , e j 0 for i j • For time series, this is written as: cov yt , ys cov et , es 0 for t s – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 10 9.1 Introduction 9.1.2a Stationarity A stationary variable is one that is not explosive, nor trending, and nor wandering aimlessly without returning to its mean Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 11 9.1 Introduction FIGURE 9.2 (a) Time series of a stationary variable 9.1.2a Stationarity Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 12 9.1 Introduction FIGURE 9.2 (b) time series of a nonstationary variable that is ‘‘slow-turning’’ or ‘‘wandering’’ 9.1.2a Stationarity Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 13 9.1 Introduction FIGURE 9.2 (c) time series of a nonstationary variable that ‘‘trends” 9.1.2a Stationarity Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 14 9.1 Introduction FIGURE 9.3 (a) Alternative paths through the chapter starting with finite distributed lags 9.1.3 Alternative Paths Through the Chapter Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 15 9.1 Introduction FIGURE 9.3 (b) Alternative paths through the chapter starting with serial correlation 9.1.3 Alternative Paths Through the Chapter Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 16 9.2 Finite Distributed Lags Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 17 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 Eq. 9.5 yt 0 xt 1 xt 1 2 xt 2 q xt q et – Note the notation change: βs is used to denote the coefficient of xt-s and α is introduced to denote the intercept Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 18 9.2 Finite Distributed Lags Model 9.5 has two uses: – Forecasting Eq. 9.6 yT 1 0 xT 1 1 xT 2 xT 1 q xT q1 eT 1 – Policy analysis • What is the effect of a change in x on y? Eq. 9.7 Principles of Econometrics, 4th Edition E ( yt ) E ( yt s ) s xt s xt Chapter 9: Regression with Time Series Data: Stationary Variables Page 19 9.2 Finite 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 q have elapsed β s s 0 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 20 9.2 Finite 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 21 9.2 Finite Distributed Lags ASSUMPTIONS OF THE DISTRIBUTED LAG MODEL 9.2.1 Assumptions TSMR1. yt β0 xt β1xt 1 β2 xt 2 βq xt q et , t q 1, , T TSMR2. y and x are stationary random variables, and et is independent of current, past and future values of x. TSMR3. E(et) = 0 TSMR4. var(et) = σ2 TSMR5. cov(et, es) = 0 t ≠ s TSMR6. et ~ N(0, σ2) Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 22 9.2 Finite Distributed Lags 9.2.2 An Example: Okun’s Law 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: Ut Ut 1 Gt GN Eq. 9.8 – We can rewrite this as: DUt β0Gt et Eq. 9.9 where DU = ΔU = Ut - Ut-1, β0 = -γ, and α = γGN Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 23 9.2 Finite Distributed Lags 9.2.2 An Example: Okun’s Law We can expand this to include lags: Eq. 9.10 DUt β0Gt β1Gt 1 β2Gt 2 βqGt q et We can calculate the growth in output, G, as: Eq. 9.11 Principles of Econometrics, 4th Edition GDPt GDPt 1 Gt 100 GDPt 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 24 9.2 Finite Distributed Lags FIGURE 9.4 (a) Time series for the change in the U.S. unemployment rate: 1985Q3 to 2009Q3 9.2.2 An Example: Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 25 9.2 Finite Distributed Lags FIGURE 9.4 (b) Time series for U.S. GDP growth: 1985Q2 to 2009Q3 9.2.2 An Example: Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 26 9.2 Finite Distributed Lags Table 9.1 Spreadsheet of Observations for Distributed Lag Model 9.2.2 An Example: Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 27 9.2 Finite Distributed Lags Table 9.2 Estimates for Okun’s Law Finite Distributed Lag Model 9.2.2 An Example: Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 28 9.3 Serial Correlation Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 29 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 30 9.3 Serial Correlation FIGURE 9.5 Scatter diagram for Gt and Gt-1 9.3.1 Serial Correlation in Output Growth Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 31 9.3 Serial Correlation 9.3.1a Computing Autocorrelation Recall that the population correlation between two variables x and y is given by: ρ xy Principles of Econometrics, 4th Edition cov x, y var x var y Chapter 9: Regression with Time Series Data: Stationary Variables Page 32 9.3 Serial Correlation 9.3.1a Computing Autocorrelation Eq. 9.12 For the Okun’s Law problem, we have: ρ1 cov Gt , Gt 1 var Gt var Gt 1 cov Gt , Gt 1 var Gt 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 33 9.3 Serial Correlation 9.3.1a Computing Autocorrelation The first-order sample autocorrelation for G is obtained from Eq. 9.12 using the estimates: 1 T cov Gt , Gt 1 Gt G Gt 1 G T 1 t 2 2 1 T var Gt Gt G T 1 t 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 34 9.3 Serial Correlation 9.3.1a Computing Autocorrelation Making the substitutions, we get: G G G T Eq. 9.13 r1 t 2 G G T t 1 Principles of Econometrics, 4th Edition t 1 t G 2 t Chapter 9: Regression with Time Series Data: Stationary Variables Page 35 9.3 Serial Correlation 9.3.1a Computing Autocorrelation Eq. 9.14 More generally, the k-th order sample autocorrelation for a series y that gives the correlation between observations that are k periods apart is: T yt y yt k y rk t k 1 T 2 yt y t 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 36 9.3 Serial Correlation 9.3.1a Computing Autocorrelation 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: Eq. 9.15 Principles of Econometrics, 4th Edition 1 T yt y yt k y T k t k 1 rk 1 T 2 yt y T t 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 37 9.3 Serial Correlation 9.3.1a Computing Autocorrelation Applying this to our problem, we get for the first four autocorrelations: Eq. 9.16 r1 0.494 r2 0.411 r3 0.154 r4 0.200 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 38 9.3 Serial Correlation 9.3.1a Computing Autocorrelation How do we test whether an autocorrelation is significantly different from zero? – The null hypothesis is H0: ρk = 0 – A suitable test statistic is: Eq. 9.17 Principles of Econometrics, 4th Edition rk 0 Z T rk 1T Chapter 9: Regression with Time Series Data: Stationary Variables N 0,1 Page 39 9.3 Serial Correlation 9.3.1a Computing Autocorrelation For our problem, we have: Z1 98 0.494 4.89, Z 2 98 0.414 4.10 Z 3 98 0.154 1.52, Z 4 98 0.200 1.98 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 40 9.3 Serial Correlation 9.3.1b The Correlagram 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 41 9.3 Serial Correlation FIGURE 9.6 Correlogram for G 9.3.1b The Correlagram Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 42 9.3 Serial Correlation 9.3.2 Serially Correlated Errors The correlogram can also be used to check whether the multiple regression assumption cov(et, es) = 0 for t ≠ s is violated Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 43 9.3 Serial Correlation 9.3.2a A Phillips Curve Consider a model for a Phillips Curve: INFt INFt E γ Ut Ut 1 Eq. 9.18 – If we initially assume that inflationary expectations are constant over time (β1 = INFEt) set β2= -γ, and add an error term: Eq. 9.19 Principles of Econometrics, 4th Edition INFt β1 β2 DUt et Chapter 9: Regression with Time Series Data: Stationary Variables Page 44 9.3 Serial Correlation FIGURE 9.7 (a) Time series for Australian price inflation 9.3.2a A Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 45 9.3 Serial Correlation FIGURE 9.7 (b) Time series for the quarterly change in the Australian unemployment rate 9.3.2a A Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 46 9.3 Serial Correlation 9.3.2a A Phillips Curve To determine if the errors are serially correlated, we compute the least squares residuals: Eq. 9.20 Principles of Econometrics, 4th Edition eˆt INFt b1 b2 DUt Chapter 9: Regression with Time Series Data: Stationary Variables Page 47 9.3 Serial Correlation FIGURE 9.8 Correlogram for residuals from least-squares estimated Phillips curve 9.3.2a A Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 48 9.3 Serial Correlation 9.3.2a A Phillips Curve The k-th order autocorrelation for the residuals can be written as: T rk Eq. 9.21 eˆ eˆ t k 1 T t t k 2 ˆ e t t 1 – The least squares equation is: INF 0.7776 0.5279 DU Eq. 9.22 Principles of Econometrics, 4th Edition se 0.0658 0.2294 Chapter 9: Regression with Time Series Data: Stationary Variables Page 49 9.3 Serial Correlation 9.3.2a A Phillips Curve The values at the first five lags are: r1 0.549 r2 0.456 r3 0.433 r4 0.420 r5 0.339 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 50 9.4 Other Tests for Serially Correlated Errors Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 51 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test An advantage of this test is that it readily generalizes to a joint test of correlations at more than one lag Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 52 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test If et and et-1 are correlated, then one way to model the relationship between them is to write: et ρet 1 vt Eq. 9.23 – We can substitute this into a simple regression equation: Eq. 9.24 Principles of Econometrics, 4th Edition yt β1 β2 xt ρet 1 vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 53 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test We have one complication: eˆ0 is unknown – Two ways to handle this are: 1. Delete the first observation and use a total of T observations 2. Set eˆ0 0 and use all T observations Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 54 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test For the Phillips Curve: i t 6.219 ii t 6.202 F 38.67 p -value 0.000 F 38.47 p -value 0.000 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 55 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test To derive the relevant auxiliary regression for the autocorrelation LM test, we write the test equation as: yt β1 β2 xt ρeˆt 1 vt Eq. 9.25 – But since we know that yt b1 b2 xt eˆt , we get: b1 b2 xt eˆt β1 β2 xt ρeˆt 1 vt Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 56 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test Rearranging, we get: eˆt β1 b1 β 2 b2 xt ρeˆt 1 vt Eq. 9.26 γ1 γ 2 xt ρeˆt 1 v – 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 goodnessof-fit statistic, respectively, from least squares estimation of Eq. 9.26 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 57 9.4 Other Tests for Serially Correlated Errors 9.4.1 A Lagrange Multiplier Test Considering the two alternative ways to handle eˆ0 : 2 iii LM T 1 R 89 0.3102 27.61 iv LM T R 90 0.3066 27.59 2 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 58 9.4 Other Tests for Serially Correlated Errors 9.4.1a Testing Correlation at Longer Lags For a four-period lag, we obtain: iii iv LM T 4 R 2 86 0.3882 33.4 LM T R 2 90 0.4075 36.7 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 59 9.4 Other Tests for Serially Correlated Errors 9.4.2 The DurbinWatson Test 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 60 9.5 Estimation with Serially Correlated Errors Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 61 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 et ρet 1 vt 3. A general estimation strategy for estimating models with serially correlated errors Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 62 9.5 Estimation with Serially Correlated Errors We will encounter models with a lagged dependent variable, such as: yt δ θ1 yt 1 δ0 xt δ1 xt 1 vt Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 63 9.5 Estimation with Serially Correlated Errors ASSUMPTION FOR MODELS WITH A LAGGED DEPENDENT VARIABLE TSMR2A In the multiple regression model yt β1 β2 xt 2 βK xK vt Where some of the xtk may be lagged values of y, vt is uncorrelated with all xtk and their past values. Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 64 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 65 9.5 Estimation with Serially Correlated Errors 9.5.1 Least 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 66 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation Consider the model yt = β1 + β2xt + et – The variance of b2 is: var b2 wt2 var et wt ws cov et , es t t s wt ws cov et , es ts wt2 var et 1 2 w t t var et t Eq. 9.27 where wt xt x Principles of Econometrics, 4th Edition x x t Chapter 9: Regression with Time Series Data: Stationary Variables 2 t Page 67 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation When the errors are not correlated, cov(et, es) = 0, and the term in square brackets is equal to one. – The resulting expression var b2 t wt2 var et is the one used to find heteroskedasticityconsistent (HC) standard errors – When the errors are correlated, the term in square brackets is estimated to obtain HAC standard errors Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 68 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation If we call the quantity in square brackets g and its estimate gˆ , then the relationship between the two estimated variances is: Eq. 9.28 Principles of Econometrics, 4th Edition varHAC b2 varHC b2 gˆ Chapter 9: Regression with Time Series Data: Stationary Variables Page 69 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation Let’s reconsider the Phillips Curve model: INF 0.7776 0.5279 DU Eq. 9.29 Principles of Econometrics, 4th Edition 0.0658 0.2294 0.1030 0.3127 Chapter 9: Regression with Time Series Data: Stationary Variables incorrect se HAC se Page 70 9.5 Estimation with Serially Correlated Errors 9.5.1 Least Squares Estimation The t and p-values for testing H0: β2 = 0 are: t 0.5279 0.2294 2.301 p 0.0238 t 0.5279 0.3127 1.688 p 0.0950 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables from LS standard errors from HAC standard errors Page 71 9.5 Estimation with Serially Correlated Errors 9.5.2 Estimating an AR(1) Error Model Return to the Lagrange multiplier test for serially correlated errors where we used the equation: et ρet 1 vt Eq. 9.30 – Assume the vt are uncorrelated random errors with zero mean and constant variances: Eq. 9.31 E vt 0 Principles of Econometrics, 4th Edition var vt v2 cov vt , vs 0 for t s Chapter 9: Regression with Time Series Data: Stationary Variables Page 72 9.5 Estimation with Serially Correlated Errors 9.5.2 Estimating an AR(1) Error Model 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-handside variable is et lagged one period Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 73 9.5 Estimation with Serially Correlated Errors 9.5.2a Properties of an AR(1) Error We assume that: 1 ρ 1 Eq. 9.32 The mean and variance of et are: Eq. 9.33 E et 0 var et 2 e 2 v 1 ρ2 The covariance term is: ρ cov et , et k , k 0 1 ρ k Eq. 9.34 Principles of Econometrics, 4th Edition 2 v 2 Chapter 9: Regression with Time Series Data: Stationary Variables Page 74 9.5 Estimation with Serially Correlated Errors 9.5.2a Properties of an AR(1) Error The correlation implied by the covariance is: ρ k corr et , et k cov et , et k Eq. 9.35 var et var et k cov et , et k var et ρ k 2 v 2 v 1 ρ 1 ρ 2 2 ρk Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 75 9.5 Estimation with Serially Correlated Errors 9.5.2a Properties of an AR(1) Error Setting k = 1: ρ1 corr et , et 1 ρ Eq. 9.36 – 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) Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 76 9.5 Estimation with Serially Correlated Errors 9.5.2a Properties of an AR(1) Error Each et depends on all past values of the errors vt: et vt ρvt 1 ρ vt 2 ρ vt 3 2 Eq. 9.37 3 – For the Phillips Curve, we find for the first five lags: r1 0.549 r2 0.456 r3 0.433 r4 0.420 r5 0.339 – For an AR(1) model, we have: ρˆ1 ρˆ r1 0.549 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 77 9.5 Estimation with Serially Correlated Errors 9.5.2a Properties of an AR(1) Error For longer lags, we have: ρˆ 2 ρˆ 0.549 0.301 2 2 ρˆ 3 ρˆ 3 0.549 0.165 3 ρˆ 4 ρˆ 4 0.549 0.091 4 ρˆ 5 ρˆ 0.549 0.050 5 Principles of Econometrics, 4th Edition 5 Chapter 9: Regression with Time Series Data: Stationary Variables Page 78 9.5 Estimation with Serially Correlated Errors 9.5.2b Nonlinear Least Squares Estimation Our model with an AR(1) error is: yt β1 β2 xt et with et ρet 1 vt Eq. 9.38 with -1 < ρ < 1 – For the vt, we have: Eq. 9.39 E vt 0 var vt v2 cov vt , vt 1 0 for t s Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 79 9.5 Estimation with Serially Correlated Errors 9.5.2b Nonlinear Least Squares Estimation With the appropriate substitutions, we get: yt β1 β2 xt ρet 1 vt Eq. 9.40 – For the previous period, the error is: et 1 yt 1 β1 β2 xt 1 Eq. 9.41 – Multiplying by ρ: Eq. 9.42 Principles of Econometrics, 4th Edition ρet 1 et yt 1 ρβ1 ρβ2 xt 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 80 9.5 Estimation with Serially Correlated Errors 9.5.2b Nonlinear Least Squares Estimation Substituting, we get: Eq. 9.43 Principles of Econometrics, 4th Edition yt β1 1 ρ β2 xt ρyt 1 ρβ2 xt 1 vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 81 9.5 Estimation with Serially Correlated Errors 9.5.2b Nonlinear Least Squares Estimation 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 82 9.5 Estimation with Serially Correlated Errors 9.5.2b Nonlinear Least Squares Estimation Eq. 9.44 Our Phillips Curve model assuming AR(1) errors is: INFt β1 1 ρ β2 DUt ρINFt 1 ρβ2 DUt 1 vt – Applying nonlinear least squares and presenting the estimates in terms of the original untransformed model, we have: INF 0.7609 0.6944 DU Eq. 9.45 se 0.1245 0.2479 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables et 0.557et 1 vt 0.090 Page 83 9.5 Estimation with Serially Correlated Errors 9.5.2c Generalized Least Squares Estimation Nonlinear least squares estimation of Eq. 9.43 is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 84 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model We have the model: yt β1 1 ρ β2 xt ρyt 1 ρβ2 xt 1 vt Eq. 9.46 – Suppose now that we consider the model: yt δ θ1 yt 1 δ0 xt δ1xt 1 vt Eq. 9.47 • 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 85 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model Note that Eq. 9.47 is the same as Eq. 9.47 since: Eq. 9.48 δ β1 1 ρ δ0 β2 δ1 ρβ2 θ1 ρ – Eq. 9.46 is a restricted version of Eq. 9.47 with the restriction δ1 = -θ1δ0 imposed Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 86 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model Applying the least squares estimator to Eq. 9.47 using the data for the Phillips curve example yields: Eq. 9.49 INF t 0.3336 0.5593INFt 1 0.6882 DU t 0.3200 DU t 1 se 0.0899 0.0908 Principles of Econometrics, 4th Edition 0.2575 Chapter 9: Regression with Time Series Data: Stationary Variables 0.2499 Page 87 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model The equivalent AR(1) estimates are: δˆ βˆ 1 1 ρˆ 0.7609 1 0.5574 0.3368 θˆ 1 ρˆ 0.5574 δˆ βˆ 0.6944 0 2 ˆ ˆ 2 0.5574 0.6944 0.3871 δˆ 1 ρβ – These are similar to our other estimates Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 88 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model The original economic model for the Phillips Curve was: INFt INFt E γ Ut Ut 1 Eq. 9.50 – Re-estimation of the model after omitting DUt-1 yields: Eq. 9.51 INF t 0.3548 0.5282 INFt 1 0.4909 DU t se 0.0876 0.0851 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables 0.1921 Page 89 9.5 Estimation with Serially Correlated Errors 9.5.3 Estimating a More General Model In this model inflationary expectations are given by: INFt E 0.3548 0.5282INFt 1 – A 1% rise in the unemployment rate leads to an approximate 0.5% fall in the inflation rate Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 90 9.5 Estimation with Serially Correlated Errors 9.5.4 Summary of Section 9.5 and Looking Ahead We have described three ways of overcoming the effect of serially correlated errors: 1. Estimate the model using least squares with HAC standard errors 2. 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 91 9.6 Autoregressive Distributed Lag Models Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 92 9.6 Autoregressive Distributed Lag Models An autoregressive distributed lag (ARDL) model is one that contains both lagged xt’s and lagged yt’s Eq. 9.52 yt 0 xt 1xt 1 q xt q 1 yt 1 p yt p vt – Two examples: ADRL 1,1 : INFt 0.3336 0.5593INFt 1 0.6882 DU t 0.3200 DU t 1 ADRL 1,0 : INFt 0.3548 0.5282 INFt 1 0.4909 DU t Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 93 9.6 Autoregressive Distributed Lag Models An ARDL model can be transformed into one with only lagged x’s which go back into the infinite past: yt 0 xt β1 xt 1 β 2 xt 2 β3 xt 3 Eq. 9.53 et β s xt s et s 0 – This model is called an infinite distributed lag model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 94 9.6 Autoregressive Distributed Lag Models 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? Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 95 9.6 Autoregressive Distributed Lag Models The Akaike information criterion (AIC) is: SSE 2 K AIC ln T T Eq. 9.54 where K = p + q + 2 The Schwarz criterion (SC), also known as the Bayes information criterion (BIC), is: SSE K ln T SC ln T T Eq. 9.55 – Because Kln(T)/T > 2K/T for T ≥ 8, the SC penalizes additional lags more heavily than does the AIC Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 96 9.6 Autoregressive Distributed Lag Models 9.6.1 The Phillips Curve Consider the previously estimated ARDL(1,0) model: Eq. 9.56 INF t 0.3548 0.5282 INFt 1 0.4909 DU t , obs 90 se 0.0876 0.0851 Principles of Econometrics, 4th Edition 0.1921 Chapter 9: Regression with Time Series Data: Stationary Variables Page 97 9.6 Autoregressive Distributed Lag Models FIGURE 9.9 Correlogram for residuals from Phillips curve ARDL(1,0) model 9.6.1 The Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 98 9.6 Autoregressive Distributed Lag Models Table 9.3 p-values for LM Test for Autocorrelation 9.6.1 The Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 99 9.6 Autoregressive Distributed Lag Models 9.6.1 The Phillips Curve For an ARDL(4,0) version of the model: INF t 0.1001 0.2354 INFt 1 0.1213INFt 2 0.1677 INFt 3 Eq. 9.57 se 0.0983 0.1016 0.1038 0.1050 0.2819INFt -4 0.7902DU t 0.1014 Principles of Econometrics, 4th Edition 0.1885 Chapter 9: Regression with Time Series Data: Stationary Variables obs 87 Page 100 9.6 Autoregressive Distributed Lag Models 9.6.1 The Phillips Curve Inflationary expectations are given by: INFt E 0.1001 0.2354INFt 1 0.1213INFt 2 0.1677INFt 3 0.2819INFt -4 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 101 9.6 Autoregressive Distributed Lag Models Table 9.4 AIC and SC Values for Phillips Curve ARDL Models 9.6.1 The Phillips Curve Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 102 9.6 Autoregressive Distributed Lag Models 9.6.2 Okun’s Law Recall the model for Okun’s Law: Eq. 9.58 DU t 0.5836 0.2020Gt 0.1653Gt 1 0.0700G t 2 , obs 96 se 0.0472 0.0324 0.0335 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables 0.0331 Page 103 9.6 Autoregressive Distributed Lag Models FIGURE 9.10 Correlogram for residuals from Okun’s law ARDL(0,2) model 9.6.2 Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 104 9.6 Autoregressive Distributed Lag Models Table 9.5 AIC and SC Values for Okun’s Law ARDL Models 9.6.2 Okun’s Law Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 105 9.6 Autoregressive Distributed Lag Models 9.6.2 Okun’s Law Now consider this version: Eq. 9.59 DU t 0.3780 0.3501DU t 1 0.1841Gt 0.0992G t 1 , obs 96 se 0.0578 0.0846 Principles of Econometrics, 4th Edition 0.0307 0.0368 Chapter 9: Regression with Time Series Data: Stationary Variables Page 106 9.6 Autoregressive Distributed Lag Models 9.6.3 Autoregressive Models An autoregressive model of order p, denoted AR(p), is given by: Eq. 9.60 yt δ θ1 yt 1 θ2 yt 2 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables θ p yt p vt Page 107 9.6 Autoregressive Distributed Lag Models 9.6.3 Autoregressive Models Consider a model for growth in real GDP: Eq. 9.61 G t 0.4657 0.3770Gt 1 0.2462Gt 2 se 0.1433 0.1000 0.1029 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables obs = 96 Page 108 9.6 Autoregressive Distributed Lag Models FIGURE 9.11 Correlogram for residuals from AR(2) model for GDP growth 9.6.3 Autoregressive Models Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 109 9.6 Autoregressive Distributed Lag Models Table 9.6 AIC and SC Values for AR Model of Growth in U.S. GDP 9.6.3 Autoregressive Models Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 110 9.7 Forecasting Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 111 9.7 Forecasting We consider forecasting using three different models: 1. AR model 2. ARDL model 3. Exponential smoothing model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 112 9.7 Forecasting 9.7.1 Forecasting with an AR Model Consider an AR(2) model for real GDP growth: Gt δ θ1Gt 1 θ2Gt 2 vt Eq. 9.62 The model to forecast GT+1 is: GT 1 δ θ1GT θ2GT 1 vT 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 113 9.7 Forecasting 9.7.1 Forecasting with an AR Model The growth values for the two most recent quarters are: GT = G2009Q3 = 0.8 GT-1 = G2009Q2 = -0.2 The forecast for G2009Q4 is: GˆT 1 δˆ θˆ 1GT θˆ 2GT 1 Eq. 9.63 0.46573 0.37700 0.8 0.24624 0.2 0.7181 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 114 9.7 Forecasting 9.7.1 Forecasting with an AR Model For two quarters ahead, the forecast for G2010Q1 is: GˆT 2 δˆ θˆ1GT 1 θˆ 2GT 0.46573 0.37700 0.71808 0.24624 0.8 Eq. 9.64 0.9334 For three periods out, it is: GˆT 3 δˆ θˆ1GT 2 θˆ 2GT 1 Eq. 9.65 0.46573 0.37700 0.93343 0.24624 0.71808 0.9945 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 115 9.7 Forecasting 9.7.1 Forecasting with an AR Model Summarizing our forecasts: – Real GDP growth rates for 2009Q4, 2010Q1, and 2010Q2 are approximately 0.72%, 0.93%, and 0.99%, respectively Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 116 9.7 Forecasting 9.7.1 Forecasting with an AR Model A 95% interval forecast for j periods into the future is given by: GˆT j t 0.975,df σˆ j where σˆ j is the standard error of the forecast error and df is the number of degrees of freedom in the estimation of the AR model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 117 9.7 Forecasting 9.7.1 Forecasting with an AR Model The first forecast error, occurring at time T+1, is: u1 GT 1 GˆT 1 δ δˆ θ1 θˆ1 GT θ2 θˆ 2 GT 1 vT 1 Ignoring the error from estimating the coefficients, we get: Eq. 9.66 Principles of Econometrics, 4th Edition u1 vT 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 118 9.7 Forecasting 9.7.1 Forecasting with an AR Model The forecast error for two periods ahead is: Eq. 9.67 u2 θ1 GT 1 GˆT 1 vT 2 θ1u1 vT 2 θ1vT 1 vT 2 The forecast error for three periods ahead is: Eq. 9.68 u3 θ1u2 θ 2u1 vT 3 θ12 θ 2 vT 1 θ1vT 2 vT 3 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 119 9.7 Forecasting 9.7.1 Forecasting with an AR Model Because the vt’s are uncorrelated with constant variance v2, we can show that: σ12 var u1 σ v2 σ 22 var u2 σ v2 1 θ12 σ var u3 σ 2 3 Principles of Econometrics, 4th Edition 2 v θ θ2 2 1 Chapter 9: Regression with Time Series Data: Stationary Variables 2 θ12 1 Page 120 9.7 Forecasting Table 9.7 Forecasts and Forecast Intervals for GDP Growth 9.7.1 Forecasting with an AR Model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 121 9.7 Forecasting 9.7.2 Forecasting with an ARDL Model Consider forecasting future unemployment using the Okun’s Law ARDL(1,1): DUt δ θ1DUt 1 δ0Gt δ1Gt 1 vt Eq. 9.69 The value of DU in the first post-sample quarter is: Eq. 9.70 DUT 1 δ θ1DUT δ0GT 1 δ1GT vT 1 – But we need a value for GT+1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 122 9.7 Forecasting 9.7.2 Forecasting with an ARDL Model Now consider the change in unemployment – Rewrite Eq. 9.70 as: UT 1 UT δ θ1 UT UT 1 δ0GT 1 δ1GT vT 1 – Rearranging: Eq. 9.71 U T 1 δ θ1 1U T θ1U T 1 δ0GT 1 δ1GT vT 1 Principles of Econometrics, 4th Edition δ θ1*U T θ*2U T 1 δ0GT 1 δ1GT vT 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 123 9.7 Forecasting 9.7.2 Forecasting with an ARDL Model 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 124 9.7 Forecasting 9.7.3 Exponential Smoothing 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 125 9.7 Forecasting 9.7.3 Exponential Smoothing One possible forecasting method is to use the average of past information, such as: yT yT 1 yT 2 yˆT 1 3 – This forecasting rule is an example of a simple (equally-weighted) moving average model with k=3 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 126 9.7 Forecasting 9.7.3 Exponential Smoothing Now consider a form in which the weights decline exponentially as the observations get older: Eq. 9.72 yˆT 1 αyT α 1 α yT 1 α 1 α yT 2 1 2 – We assume that 0 < α < 1 – Also, it can be shown that: Principles of Econometrics, 4th Edition s 0 α 1 α 1 Chapter 9: Regression with Time Series Data: Stationary Variables s Page 127 9.7 Forecasting 9.7.3 Exponential Smoothing For forecasting, recognize that: Eq. 9.73 1 α yˆT α 1 α yT 1 α 1 α 2 yT 2 α 1 α yT 3 3 – We can simplify to: Eq. 9.74 Principles of Econometrics, 4th Edition yˆT 1 αyT 1 α yˆT Chapter 9: Regression with Time Series Data: Stationary Variables Page 128 9.7 Forecasting 9.7.3 Exponential Smoothing 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: yˆt αyt 1 1 α yˆt 1 t 2,3, , T Eq. 9.75 • Choosing α that minimizes the sum of squares of the one-step forecast errors: Eq. 9.76 Principles of Econometrics, 4th Edition vt yt yˆt yt αyt 1 + 1 α yˆt 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 129 9.7 Forecasting FIGURE 9.12 (a) Exponentially smoothed forecasts for GDP growth with α = 0.38 9.7.3 Exponential Smoothing Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 130 9.7 Forecasting FIGURE 9.12 (b) Exponentially smoothed forecasts for GDP growth with α = 0.8 9.7.3 Exponential Smoothing Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 131 9.7 Forecasting 9.7.3 Exponential Smoothing The forecasts for 2009Q4 from each value of α are: α 0.38 : GˆT 1 αGT 1 α GˆT 0.38 0.8 1 0.38 0.403921 = 0.0536 α 0.8 : GˆT 1 αGT 1 α GˆT 0.8 0.8 1 0.8 0.393578 = 0.5613 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 132 9.8 Multiplier Analysis Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 133 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 134 9.8 Multiplier Analysis Let’s find multipliers for an ARDL model of the form: Eq. 9.77 yt 1 yt 1 p yt p 0 xt 1 xt 1 q xt q vt – We can transform this into an infinite distributed lag model: Eq. 9.78 yt α β0 x t + β1 xt 1 β2 xt 2 β3 xt 3 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables et Page 135 9.8 Multiplier Analysis The multipliers are defined as: yt βs s period delay multiplier xt s s β j 0 j s period interim multiplier j total multiplier β j 0 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 136 9.8 Multiplier Analysis The lag operator is defined as: Lyt yt 1 – Lagging twice, we have: L Lyt Lyt 1 yt 2 – Or: L2 yt yt 2 – More generally, we have: Ls yt yt s Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 137 9.8 Multiplier Analysis Now rewrite our model as: yt 1 Lyt 2 L2 yt p Lp yt 0 xt 1Lxt 2 L2 xt Eq. 9.79 Principles of Econometrics, 4th Edition q Lq xt vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 138 9.8 Multiplier Analysis Rearranging terms: Eq. 9.80 2 1 L L 1 2 Principles of Econometrics, 4th Edition p Lp yt 0 1L 2 L2 Chapter 9: Regression with Time Series Data: Stationary Variables q Lq xt vt Page 139 9.8 Multiplier Analysis Let’s apply this to our Okun’s Law model – The model: DUt δ θ1DUt 1 δ0Gt δ1Gt 1 vt Eq. 9.81 can be rewritten as: Eq. 9.82 Principles of Econometrics, 4th Edition 1 θ1L DUt δ δ0 δ1L Gt vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 140 9.8 Multiplier Analysis Define the inverse of (1 – θ1L) as (1 – θ1L)-1 such that: 1 θ1L 1 θ1L 1 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 141 9.8 Multiplier Analysis Multiply both sides of Eq. 9.82 by (1 – θ1L)-1: DU t 1 θ1 L δ 1 θ1 L 1 Eq. 9.83 1 δ0 δ1L Gt 1 θ1L 1 – Equating this with the infinite distributed lag representation: DU t α β0Gt β1Gt 1 β 2Gt 2 β3Gt 3 Eq. 9.84 Principles of Econometrics, 4th Edition α β0 β1L β 2 L2 β3 L3 Chapter 9: Regression with Time Series Data: Stationary Variables G e t et t Page 142 vt 9.8 Multiplier Analysis For Eqs. 9.83 and 9.84 to be identical, it must be true that: Eq. 9.85 α= 1 θ1 L δ Eq. 9.86 β0 β1L β 2 L β3 L Eq. 9.87 et 1 θ1L vt 1 2 Principles of Econometrics, 4th Edition 3 1 θ1L 1 δ0 δ1L 1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 143 9.8 Multiplier Analysis 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: δ 1 θ1 α δ and α 1 θ1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 144 9.8 Multiplier Analysis Multiply both sides of Eq. 9.86 by (1 – θ1L): δ0 δ1 L 1 θ1 L β 0 β1 L β 2 L2 β3 L3 Eq. 9.88 β0 β1 L β 2 L2 β3 L3 β0θ1 L β1θ1 L2 β 2θ1 L3 β0 β1 β0θ1 L β 2 β1θ1 L2 β3 β 2θ1 L3 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 145 9.8 Multiplier Analysis Rewrite Eq. 9.86 as: Eq. 9.89 δ0 δ1L 0L2 0L3 β0 β1 β0θ1 L β2 β1θ1 L2 β3 β2θ1 L3 – Equating coefficients of like powers in L yields: δ0 = β 0 δ1 β1 β0θ1 0 β 2 β1θ1 0 β3 β 2θ1 and so on Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 146 9.8 Multiplier Analysis We can now find the β’s using the recursive equations: β 0 = δ0 Eq. 9.90 β1 δ1 β0θ1 β j β j 1θ1 for j 2 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 147 9.8 Multiplier Analysis You can start from the equivalent of Eq. 9.88 which, in its general form, is: δ0 δ1L δ2 L2 Eq. 9.91 δq Lq 1 θ1L θ2 L2 θ p Lp β0 β1L β2 L2 β3 L3 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 148 9.8 Multiplier Analysis For the Okun’s Law model: DU t 0.3780 0.3501DUt 1 0.1841Gt 0.0992Gt 1 – The impact and delay multipliers for the first four quarters are: βˆ 0 = δˆ 0 0.1841 βˆ δˆ βˆ θˆ 0.099155 0.184084 0.350116 0.1636 1 1 0 1 βˆ 2 βˆ 1θˆ 1 0.163606 0.350166 0.0573 βˆ βˆ θˆ 0.057281 0.350166 0.0201 3 2 1 βˆ 4 βˆ 3θˆ 1 0.020055 0.350166 0.0070 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 149 9.8 Multiplier Analysis FIGURE 9.13 Delay multipliers from Okun’s law ARDL(1,1) model Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 150 9.8 Multiplier Analysis We can estimate the total multiplier given by: β j 0 j and the normal growth rate that is needed to maintain a constant rate of unemployment: GN α Principles of Econometrics, 4th Edition β j 0 Chapter 9: Regression with Time Series Data: Stationary Variables j Page 151 9.8 Multiplier Analysis We can show that: ˆ δˆ θˆ δ 0.163606 1 0 1 ˆ β j δ0 0.184084 0.4358 1 0.350116 1 θˆ j 0 1 – An estimate for α is given by: δˆ 0.37801 αˆ 0.5817 1 θˆ 0.649884 1 – Therefore, normal growth rate is: 0.5817 ˆ GN 1.3% per quarter 0.4358 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 152 Key Words Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 153 Keywords AIC criterion AR(1) error AR(p) model ARDL(p,q) model autocorrelation Autoregressive distributed lags autoregressive error autoregressive model BIC criterion correlogram delay multiplier distributed lag weight Principles of Econometrics, 4th Edition dynamic models exponential smoothing finite distributed lag forecast error forecast intervals forecasting HAC standard errors impact multiplier infinite distributed lag interim multiplier lag length lag operator lagged dependent variable Chapter 9: Regression with Time Series Data: Stationary Variables LM test multiplier analysis nonlinear least squares out-of-sample forecasts sample autocorrelations serial correlation standard error of forecast error SC criterion total multiplier T x R2 form of LM test within-sample forecasts Page 154 Appendices Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 155 9A The DurbinWatson Test For the Durbin-Watson test, the hypotheses are: H0 : 0 H1 : 0 The test statistic is: T Eq. 9A.1 d eˆt eˆt 1 2 t 2 T 2 ˆ e t t 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 156 9A The DurbinWatson Test We can expand the test statistic as: d T T T t 2 t 2 t 2 2 2 ˆ ˆ e e t t 1 2 eˆt eˆt 1 T 2 ˆ e t t 1 T Eq. 9A.2 eˆ t 2 T 2 t 2 ˆ e t t 1 T eˆ t 2 T 2 t 1 2 ˆ e t T 2 t 1 eˆt eˆt 1 t 2 T 2 ˆ e t t 1 1 1 2r1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 157 9A The DurbinWatson Test We can now write: d 2 1 r1 Eq. 9A.3 – 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 158 9A The DurbinWatson Test Principles of Econometrics, 4th Edition FIGURE 9A.1 Testing for positive autocorrelation Chapter 9: Regression with Time Series Data: Stationary Variables Page 159 9A The DurbinWatson Test FIGURE 9A.2 Upper and lower critical value bounds for the DurbinWatson test 9A.1 The DurbinWatson Bounds Test Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 160 9A The DurbinWatson Test 9A.1 The DurbinWatson Bounds Test Decision rules, known collectively as the DurbinWatson 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 161 9B Properties of the AR(1) Error Note that: et ρet 1 vt ρ ρet 2 vt 1 vt Eq. 9B.1 ρ2 et 2 ρvt 1 vt Further substitution shows that: Eq. 9B.2 Principles of Econometrics, 4th Edition et ρ2 ρet 3 vt 2 ρvt 1 vt ρ3et 3 ρ2vt 2 ρvt 1 vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 162 9B Properties of the AR(1) Error Repeating the substitution k times and rearranging: Eq. 9B.3 et ρ et k vt ρvt 1 ρ vt 2 k 2 k 1 ρ vt k 1 If we let k → ∞, then we have: Eq. 9B.4 Principles of Econometrics, 4th Edition et vt ρvt 1 ρ2vt 2 ρ3vt 3 Chapter 9: Regression with Time Series Data: Stationary Variables Page 163 9B Properties of the AR(1) Error We can now find the properties of et: E et E vt ρE vt 1 ρ 2 E vt 2 ρ3 E vt 3 0 ρ 0 ρ 2 0 ρ3 0 0 var et var vt ρ 2 var vt 1 ρ 4 var vt 2 ρ6 var vt 3 v2 ρ 2 v2 ρ 4 v2 ρ6 v2 v2 1 ρ 2 ρ 4 ρ6 v2 1 ρ2 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 164 9B Properties of the AR(1) Error The covariance for one period apart is: cov et , et 1 E et et t 2 3 E vt ρvt 1 ρ vt 2 ρ vt 3 v ρv ρ v ρ v ρE v ρ E v ρ E v ρ 1 ρ ρ t 1 t 2 2 t 1 3 2 v 2 2 2 t 2 3 t 3 5 t 4 2 t 3 4 ρv2 1 ρ2 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 165 9B Properties of the AR(1) Error Similarly, the covariance for k periods apart is: ρ k v2 cov et , et k 1 ρ2 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables k 0 Page 166 9C Generalized Least Squares Estimation We are considering the simple regression model with AR(1) errors: yt 1 2 xt et et et 1 vt To specify the transformed model we begin with: yt 1 2 xt yt 1 1 2 xt 1 vt Eq. 9C.1 – Rearranging terms: Eq. 9C.2 Principles of Econometrics, 4th Edition yt yt 1 1 1 2 xt xt 1 vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 167 9C Generalized Least Squares Estimation Defining the following transformed variables: yt yt yt 1 xt2 xt xt 1 xt1 1 Substituting the transformed variables, we get: Eq. 9C.3 Principles of Econometrics, 4th Edition yt xt11 xt22 vt Chapter 9: Regression with Time Series Data: Stationary Variables Page 168 9C Generalized Least Squares Estimation 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 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 169 9C Generalized Least Squares Estimation For the second problem, we can write Eq. 9C.1 as: yt 1 2 xt ( yt 1 1 2 xt 1 ) vt Eq. 9C.4 For the first problem, note that: y1 1 x12 e1 and that 1 2 y1 1 2 1 1 2 x12 1 2 e1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 170 9C Generalized Least Squares Estimation Or: y1 x11 1 x12 2 e1 Eq. 9C.5 where y1 1 2 y1 x11 1 2 x12 1 2 x1 e1 1 2 e1 Eq. 9C.6 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 171 9C Generalized Least Squares Estimation To confirm that the variance of e*1 is the same as that of the errors (v2, v3,…, vT), note that: 2 var(e1 ) (1 2 ) var(e1 ) (1 2 ) v 2 v2 1 Principles of Econometrics, 4th Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 172