Economics 140A Error: Serial Correlation
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1 Economics 140A Error: Serial Correlation We continue with the remaining classic assumptions that refer to the error and are statistical in meaning. Today we focus on the assumption that the error has zero serial correlation. The assumption of zero serial correlation is closely related to the assumption of homoskedasticity in that, violation of either assumption allows one to construct a more e cient estimator by weighting the observations unequally. Recall that if the error is heteroskedastic, then the observations are not equally accurate and should be weighted according to their accuracy. If the error is serially correlated, then knowledge of one value of the error term contains information about the other values of the error term and so can be used to improve estimation. In the classic model we assume that each observation does not provide information about the unknown error for any other observation; E (U s U t ) = 0 if s and t index di erent observations. If the assumption is satis ed, the covariance (and so, the correlation) between any two observations is zero. As the observations are often related through time, that is they are serial in nature, the assumption is often said to imply that the error is serially uncorrelated. If the assumption is violated, so that E (U s U t ) = s;t; then the error is said to be serially correlated. Why might the error be serially correlated? One natural cause would be an omitted variable. (Studenmund refers to this as impure serial correlation.) If we have incorrectly excluded X t;k from the regression, then the regression error is U t + K X t;k. If X t;k is serially correlated, then so too is the error. The example brings to light why many state that error serial correlation is largely a problem of data in which the regressors are serially correlated. The solution here is to include the omitted variable, which also eliminates serial correlation. Yet serial correlation may arise for other reasons. For example, if the mismeasurement of the dependent variable arises from the introduction of a poorly measured component (as happened when computers were rst introduced into the price index), the measurement error may be serially correlated. Incorrect functional form can also lead to serial correlation in the error. Suppose the population model is a curve from the origin and the estimated model is a line. For low values of the regressor, the population curve lies below the estimated line and the errors will tend to be negative. For high values of the regressor, the population curve lies
2 above the estimated line and the errors will tend to be positive. If the regressors are grouped by size, as would be the case if the value of the regressor grew over time, then the error is serially correlated. As this example makes clear, serial correlation may arise from model misspeci cation, and yet not be easily cured by adding in another variable. Graph with exponential and line from origin What are the e ects of serial correlation? We focus on the case in which the serial correlation is viewed as a problem of measurement of the dependent variable. As such, serial correlation in the error does not lead to correlation between the regressors and the error and so the OLS coe cient estimator remains unbiased. Although the OLS coe cient estimator is unbiased, it is ine cient. Further, the estimator of the variance of the OLS coe cient estimator is incorrect if the serial correlation is unaccounted for. The clean pedagogical solution, is to transform the model to one in which the error is serially uncorrelated. For the transformed model the OLS coe cient estimator is the e cient unbiased (linear) estimator and the estimator of the variance of the OLS coe cient estimator is correct. Consider the two-variable model in deviation-from-means form Y t = X t + U t ; where E (U s U t ) = s;t. From the de nition of the covariance between two random variables, it is clear that location and scale transformations of the entire sequence will not remove the covariance. That is, if one knew only that E (U s U t ) =, then there is no transformation of U s and U t involving such that the resultant variables have zero covariance. (If the two variables positively covary, so that when U s is above its mean, U t tends to be above its mean as well, then shifting the location of either (or both) will not change the covariance. Similarly, shifting the scale of either or both will not eliminate the covariance either.) As such, we must go further then we did for the heteroskedastic case and specify the dynamic relation that leads to the nonzero covariance. In many applications in which serial correlation is present, it is reasonable to assume a far simpler model for the serial covariance in which s;t = jt For the simpler form, only the distance between the two observations matters; it does not matter where in the sequence the two observations occur. The absolute sj:
3 value sign in the subscript arises because, by de nition, E (U s U t ) = E (U t U s ), so the value of the covariance does not depend on the order in which the two errors occur. With the dynamic relation in hand, one can undo the dynamic relation and obtain variables that have zero covariance. A model of serial correlation that is surprisingly widely applicable is U t = U t 1 + V t ; where fv t g n t=1 is a sequence of random variables that satisfy the classic assumptions. One can view the model as a regression, in which the dependent variable is regressed on a lag of the dependent variable. Such a speci cation is termed an autoregression. Although we will not stress the point in this class, the dynamic structure of an autoregression requires that we discuss the magnitude of ; here we restrict jj < 1. If the error is known to follow an autoregression and is known, then U t U t 1 = V t ; is serially uncorrelated and the transformation Y t Y t 1 = (X t X t 1 ) + (U t U t 1 ) ; (0.1) yields a serially uncorrelated error. The OLS coe cient estimator for (0.1) is often termed the -di erence (or quasi-di erence) transformation. The estimator of the variance of the -di erence estimator U t Ut 1 (X t X t 1 ) is correct. In constructing the -di erence estimator, use (0.1) only to estimate. With the estimate of in hand, return to the original model for interpretation and prediction. The -di erence estimator di ers slightly from the the generalized least squares estimator (the term general re ects a transformation that is more general than reweighting) in that it uses only n 1 observations. The GLS estimator, which is constructed using the covariance matrix for U t, also constructs an OLS estimator for (0.1) but includes (1 ) 1 Y 1 and (1 ) 1 X 1 as the observations for the rst period and so uses all n observations. Inclusion of the rst observation implies that the GLS estimator is the e cient unbiased (linear) estimator of. If n is large, the e ciency gain from including the rst term is slight. 3
4 Given knowledge that the serial correlation model is an autoregression as used in (0.1), one is left only to determine. The most e cient method is simply to estimate (0.1) directly Y t = Y t 1 + X t X t 1 + (U t U t 1 ) : One cannot simply use an OLS estimator, as the model is not linear in the coef- cients (the coe cient on X t 1 is ). Rather one must use a nonlinear least squares estimator. We nd arg min ~A; ~ B Y t ~ AYt 1 ~ BXt + ~ A ~ BX t 1 : The rst-order conditions are Y A ~ = t AYt ~ 1 BXt ~ + A ~ BX ~ ~BXt t 1 1 Y t 1 and ~ B = Y t ~ AYt 1 ~ BXt + ~ A ~ BX t 1 ~AXt 1 X t : To understand intuitively how to solve the system, recall the rst-order conditions for the OLS estimator of the two-variable regression model and ~ A = ~ B = t=1 t=1 Y t ~ A ~ BXt ( 1) Y t ~ A ~ BXt ( X t ) : If the two conditions are set equal to zero, the resultant equations are A = Y n B X n and A = t=1 Y tx t t=1 X t B t=1 X : t Consider the plane with the di erent possible values of A on the vertical axis and the di erent possible values of B on the horizontal axis. The two equations form lines in the plane with di erent slopes and so must cross. The point of crossing is the solution. 4
5 (See Figure 1) One could consult my musings here. I would simply mention that the two relations are curves. I think this issue is more for the graduate class than for the undergraduate class. For the nonlinear least squares problem, if A ~ is set equal to zero, then A = Y ty t 1 + B X tx t 1 B X ty t 1 B Y tx t 1 n Y t 1 + B n X t 1 X t 1Y t 1 B X : t 1Y t 1 where c 1 (A) = A and Xt 1; c (A) = c 3 (A) = In similar fashion, if ~ B B = B c 1 (A) + Bc (A) + c 3 (A) = 0; A X t Y t 1 + Y t 1 A Y t X t 1 X t X t 1 A! X t 1 Y t 1 Y t Y t 1 : is set equal to zero, then X ty t + A n X t 1Y t 1 A X ty t 1 A Y tx t 1 X : tx t 1 A X t 1 + X t A X tx t 1 A An alternative procedure, which is easier to explain, is the Cochrane-Orcutt procedure. In the rst step, the OLS estimator of is constructed and a regression of U t on U t 1 yields an estimator of, which we term A CO. The second step is to construct the OLS estimator of from Y t A CO Y t 1 = (X t A CO X t 1 ) + U t ; where U t = U t A CO U t 1. The procedure can be iterated. How might one know that the autoregression modeled above is correct? The use of the Lagrange multiplier test statistic is the principle method. To implement the test, rst construct the OLS residuals U t. With the OLS residuals in hand, construct the OLS coe cient estimators for U t = U t 1 + V t : 5! X t 1 Y t 1
6 (Note, if the regressors include a lagged value of the dependent variable, then the regressors must be included as well to control for correlation between the lagged dependent variable and the error.) Because Ut is orthogonal to fx t=1 tg n t=1, any explanatory power in the preceding regression must be due to Ut 1. In fact n R, where R is the estimator of R-squared from the preceding regression, is a random variable with 1 degree-of-freedom. We reject H 0 : = 0 in favor of H 1 : 6= 0 if the statistic is large enough. The Lagrange multiplier test statistic is exible. If the data is measured quarterly, so that it is most likely that U t is correlated with U t 4 (and not U t 1 ) we simply replace Ut 1 with Ut 4 in the regression. Also, if one believes that the error is correlated with the two preceding values, simply include both Ut 1 and Ut in the regression and note that the test statistic is a random variable with degrees-of-freedom. The exibility is a major advantage over the Durbin-Watson test statistic, which is still reported by some software programs. The Durbin-Watson test statistic is a direct test only of correlation between U t and U t 1 and is formed as U t U t 1 t=1 (U t ) 1! n U t 1Ut t=1 (U t ) ; where the later fraction is an estimator of the correlation between Ut and Ut 1. Unsurprsingly, values of the test statistic close to two provide little evidence of serial correlation; those signi cantly less than two provide evidence of positive serial correlation. 1 Of course, knowledge that the serial correlation is formed by an autoregression with known coe cient is rare. With less knowledge, the above transformation is infeasible. One can either assume an autoregression and estimate the weight, resulting in feasible generalized least squares, or work with original regression and obtain a correct estimator of the variance of the OLS coe cient estimator. For the case at hand, in which the regressors are exogenous, the decision about which method to follow turns on the precision with which the transformation is estimated. In some applied work, researchers believe that the transformation is estimated too imprecisely to use generalized least squares. The question then turns to correct estimation of the variance of the OLS coe cient estimator. For the two-variable model introduced above, the variance of the OLS coe cient estimator 1 An additional di culty of the Durbin-Watson test statistic is that the critical values depend on the observed regressor values. As such, there is a range of indeterminacy in which the null hypothesis can neither be rejected, nor fail to be rejected. 6
7 is V (B) = E (B ) = E t=1 X tu t n : Because we condition on the observed values of the regressor E t=1 X tu t n = = E U t + s=1 t=1 s6=t X s X t U s U t! ( ) n + s=1 t=1x s X t E (U s U t ) s6=t ( ) ; where the second line follows because the error is homoskedastic. While it is not possible to consistently estimate all the error covariances, it is possible to consistently estimate the variance of the OLS coe cient estimator. Intuitively, we do not need to know all the values of n s;t but only the scalar s=1 t=1 s6=t Xn 1 X s X t E (U s U t jx) = j=1 t=j+1 s;t=1 X t j X t E (U t j U t jx) : Unlike the case of homoskedasticity, we cannot use the entire sum. Intuitively, one may think that the problem arises because the covariances in the sum range from those that are adjacent (for which jt sj = 1) to those on opposite ends of the sample (for which jt sj = n 1). While we have many observations from which to estimate the covariance of adjacent observations, we have only one observation from which to estimate the covariance at the furthest degree of separation. Yet if we recall the case of pure heteroskedasticity, we have only one observation with which to estimate each variance. As we are able to consistently estimate the sum in the heteroskedastic case, the fact that we have only one observation for the longest lag autocovariance cannot be the problem. erhaps the easiest way to understand the di culty, is to consider the case with heteroskedasiticy and serial correlation. In this case, V ar (BjX) is s=1 t=1 X sx t E (U s U t jx) ( ) : 7
8 If we follow the logic of the Eicker-White HC variance estimator, we replace the unobserved error with the estimated residual s=1 t=1 X sx t Us Ut ( ) : The numerator of this term is written as X s Us X t Ut = 0: s=1 t=1 The equality with zero follows from the defnition of the OLS estimator (which sets the residuals orthogonal to the regressors). Hence, if we used all the covariances, the logic of Eicker-White would deliver an estimator of the variance that is always 0. Rather, the additional di culties are two fold. Our rst di culty involves the number of parameters. In the heteroskedastic case, we have n variances to n(n 1) estimate, while with serial correlation (and homoskedasticity) we have covariances to estimate. Because grows faster than n, we quickly nd that n(n 1) we have more parameters than data points (if n = we have two parameters ; ;1 while if n = 3 we have four parameters ; ;1; 3;1; 3; ). As a result, we must truncate the sum to obtain a consistent estimator. We consider terms that are at most J observations apart. The quantity J is termed the lag truncation parameter and must be estimated. Even with truncation, there is a second di culty. Unlike estimators of variances, which are always positive, estimators of covariance may be negative. As a result it is possible to obtain an estimator of the variance of the OLS coe cient estimator that is negative. To overcome the problem, we must not only truncate the sum, we must also weight the sum, which we express by rewriting the sum as J j=1 w j t=j+1 X t jx t Ut jut, with j w j = 1 a weight that declines in j. (The consistent estimator (often termed J the autocorrelation-consistent estimator) is S B;AC = S + J j=1 w j t=j+1 X t jx t Ut jut ( ) ; for which the bias is substantially less than for the uncorrected variance estimator. In many cases, the researcher allows for the possible presence of both serial correlation and heteroskedasticity S B;HAC = U t + J j=1 w j t=1 X t jx t Ut jut ( ) : 8
9 The uncorrected estimator of the variance of the OLS coe cient estimator is S B = S with S = 1 n U t ; t=1 which is a biased and inconsistent estimator of V (B). To see the direction of the bias for the case in which only serial correlation is present, note that the uncorrected estimator di ers only in that it is missing the second term in the numerator. Because the sign of serial correlation in economic data is generally positive, the omitted term is generally positive and so the uncorrected variance estimator is too small. (The sign of the omitted term is the sign of the omitted covariances.) With a downward biased variance estimator, we are over con dent of the precision with which we estimate coe cients. 9
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