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Using SAS for Time Series Data LSU Economics Department March 16, 2012 Next Workshop March 30 Instrumental Variables Estimation Time-Series Data: Nonstationary Variables Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 3 Chapter Contents 12.1 Stationary and Nonstationary Variables 12.2 Spurious Regressions 12.3 Unit Root Tests for Nonstationarity Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 4 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. Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 5 12.1 Stationary and Nonstationary Variables Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 6 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 7 12.1 Stationary and Nonstationary Variables Principles of Econometrics, 4th Edition FIGURE 12.1 U.S. economic time series Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 8 12.1 Stationary and Nonstationary Variables Principles of Econometrics, 4th Edition FIGURE 12.1 (Continued) U.S. economic time series Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 9 12.1 Stationary and Nonstationary Variables 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 10 12.1 Stationary and Nonstationary Variables That is, the time series yt is stationary if for all values, and every time period, it is true that: Eq. 12.1a E yt μ (constant mean) Eq. 12.1b var yt σ 2 (constant variance) Eq. 12.1c cov yt , yt s cov yt , yt s γ s (covariance depends on s, not t ) Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 11 12.1 Stationary and Nonstationary Variables FIGURE 12.2 Time-series models 12.1.1 The First-Order Autoregressive Model Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 12 12.1 Stationary and Nonstationary Variables FIGURE 12.2 (Continued) Time-series models 12.1.1 The First-Order Autoregressive Model Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 13 12.2 Spurious Regressions FIGURE 12.3 Time series and scatter plot of two random walk variables Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 14 12.2 Spurious Regressions A simple regression of series one (rw1) on series two (rw2) yields: rw1t 17.818 0.842 rw2t , (t ) R 2 0.70 (40.837) – These results are completely meaningless, or spurious • The apparent significance of the relationship is false Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 15 12.2 Spurious Regressions 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 16 12.3 Unit Root Tests for Stationarity Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 17 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 18 12.3 Unit Root Tests for Stationarity 12.3.1 Dickey-Fuller Test 1 (No constant and No Trend) Consider the AR(1) model: yt yt 1 vt Eq. 12.4 – We can test for nonstationarity by testing the null hypothesis that ρ = 1 against the alternative that |ρ| < 1 • Or simply ρ < 1 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 19 12.3 Unit Root Tests for Stationarity 12.3.1 Dickey-Fuller Test 1 (No constant and No Trend) A more convenient form is: yt yt 1 yt 1 yt 1 vt yt 1 yt 1 vt Eq. 12.5a yt 1 vt – The hypotheses are: H0 : 1 H0 : 0 H1 : 1 H1 : 0 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 20 12.3 Unit Root Tests for Stationarity 12.3.2 Dickey-Fuller Test 2 (With Constant but No Trend) The second Dickey–Fuller test includes a constant term in the test equation: yt yt 1 vt Eq. 12.5b – The null and alternative hypotheses are the same as before Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 21 12.3 Unit Root Tests for Stationarity 12.3.3 Dickey-Fuller Test 3 (With Constant and With Trend) The third Dickey–Fuller test includes a constant and a trend in the test equation: yt yt 1 t vt Eq. 12.5c – The null and alternative hypotheses are H0: γ = 0 and H1:γ < 0 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 22 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 23 12.3 Unit Root Tests for Stationarity Table 12.2 Critical Values for the Dickey–Fuller Test 12.3.4 The Dickey-Fuller Critical Values Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 24 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 25 12.3 Unit Root Tests for Stationarity 12.3.4 The Dickey-Fuller Critical Values An important extension of the Dickey–Fuller test allows for the possibility that the error term is autocorrelated – Consider the model: m yt yt 1 as yt s vt Eq. 12.6 where s 1 yt 1 yt 1 yt 2 , yt 2 yt 2 yt 3 , Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 26 12.3 Unit Root Tests for Stationarity 12.3.6 The Dickey-Fuller Tests: An Example 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 27 12.3 Unit Root Tests for Stationarity 12.3.6 The Dickey-Fuller Tests: An Example The results from estimating the resulting equations are: Ft 0.173 0.045 Ft 1 0.561Ft 1 (tau ) ( 2.505) Bt 0.237 0.056 Bt 1 0.237 Bt 1 (tau ) ( 2.703) – The 5% critical value for tau (τc) is -2.86 – Since -2.505 > -2.86, we do not reject the null hypothesis Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 28 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration Recall that if yt follows a random walk, then γ = 0 and the first difference of yt becomes: yt yt yt 1 vt – 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 29 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration The results of the Dickey–Fuller test for a random walk applied to the first differences are: F t 0.447 F t 1 (tau ) ( 5.487) B t 0.701 B t 1 (tau ) Principles of Econometrics, 4th Edition ( 7.662) Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 30 12.3 Unit Root Tests for Stationarity 12.3.7 Order of Integration 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) Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 31 12.4 Cointegration Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 32 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 33 12.4 Cointegration 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 34 12.4 Cointegration The test for stationarity of the residuals is based on the test equation: eˆt γeˆt 1 vt Eq. 12.7 – 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 35 12.4 Cointegration Principles of Econometrics, 4th Edition Table 12.4 Critical Values for the Cointegration Test Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 36 12.4 Cointegration There are three sets of critical values – Which set we use depends on whether the residuals are derived from: Eq. 12.8a Equation 1: eˆt yt bxt Eq. 12.8b Equation 2 : eˆt yt b2 xt b1 Eq. 12.8c Equation 3: eˆt yt b2 xt b1 ˆ t Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 37 12.4 Cointegration 12.4.1 An Example of a Cointegration Test Eq. 12.9 Consider the estimated model: Bˆt 1.140 0.914Ft , R 2 0.881 (t ) (6.548) (29.421) – The unit root test for stationarity in the estimated residuals is: eˆt 0.225eˆt 1 0.254eˆt 1 (tau ) (4.196) Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 38 12.4 Cointegration 12.4.1 An Example of a Cointegration Test The null and alternative hypotheses in the test for cointegration are: H 0 : the series are not cointegrated residuals are nonstationary H1 : the series are cointegrated residuals are stationary – 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 Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 39 Chapter 9 Regression with Time Series Data: Stationary Variables Walter R. Paczkowski Rutgers University Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 40 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 41 9.1 Introduction Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 42 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 43 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 44 9.1 Introduction Principles of Econometrics, 4th Edition FIGURE 9.1 The distributed lag effect Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 45 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 46 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 47 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 48 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 49 9.2 Finite Distributed Lags Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 50 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 51 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 52 9.3 Serial Correlation Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 53 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 54 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 55 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 56 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 57 9.3 Serial Correlation FIGURE 9.6 Correlogram for G 9.3.1b The Correlagram Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 58 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 59 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: Eq. 9.22 Principles of Econometrics, 4th Edition INF 0.7776 0.5279 DU se 0.0658 0.2294 Chapter 9: Regression with Time Series Data: Stationary Variables Page 60 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 61 9.4 Other Tests for Serially Correlated Errors Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 62 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 63 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 64 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 65 9.5 Estimation with Serially Correlated Errors Principles of Econometrics, 4th Edition Chapter 12: Regression with Time-Series Data: Nonstationary Variables Page 66 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 67 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 68 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 12: Regression with Time-Series Data: Nonstationary Variables Page 69 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 70 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 71 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 72 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 73 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 74 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 75 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 76