Autocorrelation or Serial Correlation

Transcription

1 Chapter 6 Autocorrelation or Serial Correlation

2 Section 6.1 Introduction 2

3 Evaluating Econometric Work How does an analyst know when the econometric work is completed? 3

4 4 Evaluating Econometric Work Econometric Results indeed a term that for many economists conjures up horrifying visions of well-meaning but perhaps marginally skilled, likely nocturnal, individuals sorting through endless piles of computer print-outs. One final print-out is then chosen for seemingly mysterious reasons and the rest discarded to be recycled through a local paper processor and another computer printer for other econometricians to repeat the process ad infinitum. Besides supplying a lucrative business for the paper recyclers, what useful output, if any, results from such a process? This question lies at the heart of the so called science of econometrics as currently applied, a practice which has been called data-mining, number crunching, model sifting, data grubbing, fishing, data messaging, and even alchemy among other less palatable terms. All of these euphemisms describe basically, the same process: choosing an econometric model based on repeated experimentation with available sample data. - Ziemer (1984)

5 5 Evaluating Econometric Work Econometrics may not have the everlasting charm of Holmesian characters and adventures, or even a famous resident of Baker Street, but there is much in his methodological approach to the solving of criminal cases that is of relevance to applied econometric modeling. Holmesian detection may be interpreted as accommodating the relationship between data theory, modeling procedures, deductions and inferences, analysis of biases, testing of theories, re-evaluation and reformulation of theories, and finally reaching a solution to the problem at hand. With this in mind, can applied econometricians learn anything from the master of detection? - McAleer (1994)

6 Key Diagnostics Diagnostic Serial Correlation (Autocorrelation) Heteroscedasticity Collinearity Diagnostics Influence diagnostics Structural Change Component of Econometric Model Under Consideration Error (or Disturbance) Term Error (or Disturbance) Term Explanatory Variables Observations Structural Parameters See Beggs (1988). 6

7 Section 6.2 Autocorrelation or Serial Correlation

8 8 Autocorrelation or Serial Correlation Definition Consequences Formal Tests Durbin-Watson Test Nonparametric Runs Test Durbin's h-test Durbin's m-test Lagrange Multiplier (LM) Test Box-Pierce Test (Q Statistic) Ljung-Box Test (Q* Statistic) (Small-sample modification of Box-Pierce Q Statistic) Solution Generalized Least Squares

9 Formal Definition of Autocorrelation or Serial Correlation Autocorrelation or serial correlation refers to the lack of independence of error (or disturbance) terms. Autocorrelation and serial correlation refer to the same phenomenon. Simply put, a systematic pattern exists in the residuals of the econometric model. Ideally, the residuals, which represent a composite of all factors not embedded in the model, should exhibit no pattern. That is to say, the residuals should follow a white-noise (or random) pattern. 9

10 Prevalence of Serial Correlation With the use of time-series data in econometric applications, serial correlation is public enemy number one. Systematic patterns in the error terms commonly arise due to the (inadvertent) omission of explanatory variables in econometric models. These variables might come from disciplines other than economics, finance, or business; for example, psychology and sociology. Alternatively, these variables might represent factors that simply are difficult to quantify, such as tastes and preferences of consumers or technological innovation on the part of producers. 10

11 Consequences of Serial Correlation Bishop (1981) Errors contaminated with autocorrelation or serial correlation Potential of discovering spurious relationships due to problems with autocorrelated errors (Granger and Newbold, 1974) Difficulties with structural analysis and forecasting If the error structure is autoregressive, then OLS estimates of the regression parameters are (1) unbiased, (2) consistent, but (3) inefficient in small and in large samples. 11 continued...

12 Consequences of Serial Correlation The estimates of the standard errors of the coefficients in any econometric model are biased downward if the residuals are positively autocorrelated. They are biased upward if the residuals are negatively autocorrelated. Therefore, the calculated t statistic is biased upward or downward in the opposite direction of the bias in the estimated standard error of that coefficient. Granger and Newbold (1974) further suggest that the econometric results can be defined as nonsense if R 2 >DW(d). 12 continued...

13 Consequences of Serial Correlation Positive autocorrelation of the errors generally tends to make the estimate of the error variance too small, so confidence intervals are too narrow and null hypotheses are rejected with a higher probability than the stated significance level. Negative autocorrelation of the errors generally tends to make the estimate of the error variance too large, so confidence intervals are too wide; also the power of significance tests is reduced. With either positive or negative autocorrelation, leastsquares parameter estimates usually are not as efficient as generalized least-squares parameter estimates. 13

14 Regression with Autocorrelated Errors 14 Ordinary regression analysis is based on several statistical assumptions. One key assumption is that the errors are independent of each other. However, with time series data, the ordinary regression residuals usually are correlated over time. Violation of the independent errors assumption has three important consequences for ordinary regression. First, statistical tests of the significance of the parameters and the confidence limits for the predicted values are not correct. Second, the estimates of the regression coefficients are not as efficient as they would be if the autocorrelation were taken into account. Third, because the ordinary regression residuals are not independent, they contain information that can be used to improve the prediction of future values.

15 Solution to the Serial Correlation Problem Generalized Least Squares (GLS) The AUTOREG procedure solves this problem by augmenting the regression model with an autoregressive model for the random error, thereby accounting for the systematic pattern of the errors. Instead of the usual regression model, the following autoregressive error model is used: y ε x β + ε t = t + t = φ ε 1 t t 1 φ 2 ε t 2... φ m ε t m + v t v t ~ IN(0, σ 2 ) v t ~ IN(0, σ 2 ) The notation indicates that each v t is normally and independently distributed with mean 0 and variance σ continued...

16 Solution to the Serial Correlation Problem Generalized Least Squares (GLS) By simultaneously estimating the regression coefficients β and the autoregressive error model parameters φ i, the AUTOREG procedure corrects the regression estimates for autocorrelation. Thus, this kind of regression analysis is often called autoregressive error correction or serial correlation correction. This technique is also called the use of generalized least squares (GLS). 16

17 Predicted Values and Residuals The AUTOREG procedure can produce two kinds of predicted values and corresponding residuals and confidence limits. The first kind of predicted value is obtained from only the structural part of the model. This predicted value is an estimate of the unconditional mean of the dependent variable at time t. The second kind of predicted value includes both the structural part of the model and the predicted value of the autoregressive error process. Both the structural part and autoregressive error process of the model (termed the full model) are used to forecast future values. 17 continued...

18 Predicted Values and Residuals Use the OUTPUT statement to store predicted values and residuals in a SAS data set and to output other values such as confidence limits and variance estimates. The P= option specifies an output variable to contain the full model predicted values. The PM= option names an output variable for the predicted (unconditional) mean. The R= and RM= options specify output variables for the corresponding residuals, computed as the actual value minus the predicted value. 18

19 Serial Correlation Disturbance terms are not independent. The correlation between ε t and ε t-k is called an j i E T i X X X Y j i t kt k t t t = = 0 ) ( 1,2,..., ε ε ε β β β β t t-k autocorrelation of order k continued...

20 Serial Correlation Recommend a graphical analysis of plotting the residuals over time to determine the existence of a non-random or systematic pattern. Residuals Time Positive Correlation Residuals Time Negative Correlation 20...

21 Section 6.3 Tests for Serial Correlation

22 The Durbin-Watson Test AR(1) process H H 0 1 : ρ = 0 : ρ 0 = n t= 2 0 DW If ε t (ˆ e t eˆ t n t= 1 ρε = t 1 1 ) DW ( d ) 2(1 ρ ) so that ˆ ρ = ρ = 0, then d = 2; if ρ = 1, then d ˆ 2 e t v t or d statistic DW ( d ) 1 2 = 0; n t= 2 n eˆ eˆ t t= 1 e t 1 2 t if ρ = -1, then d = 4 22 continued...

23 The Durbin-Watson Test d L,d U depend on α, k, n. DW is invalid with models that contain no intercept and models that contain lagged dependent variables. The distribution of DW(d) is reported by Durbin and Watson (1950, 1951). 23 continued...

24 f(d) Reject H 0 Reject H 0 Evidence of positive autocorrelation Zone of indecision Zone of indecision Evidence of negative autocorrelation Accept H 0 0 d d L d U 2 4-d U 4-d L 4 24 continued...

25 The Durbin-Watson Test The sampling distribution of d depends on the values of the exogenous variables and hence Durbin and Watson derived upper (d U ) limits and lower (d L ) limits for the significance levels for d. Tables of the distribution are found in most econometric textbooks. The Durbin-Watson test perhaps is the most used procedure in econometric applications. 25

26 The Durbin-Watson Statistic 26 continued...

27 27 Appendix G: Statistical Table

28 Limitations of the Durbin-Watson Test Although the Durbin-Watson test is the most commonly used test for serial correlation, there are limitations: 1. The test is for first-order serial correlation only. 2. The test might be inconclusive. 3. The test cannot be applied in models with lagged dependent variables. 4. The test cannot be applied in models without intercepts. 28

29 Additional Tables There are other tables for the DW test that have been prepared to take care of special situations. Some of these are: 1. R.W. Farebrother (1980) provides tables for regression models with no intercept term. 2. Savin and White (1977) present tables for the DW test for samples with 6 to 200 observations and for as many as 20 regressors. 29 continued...

30 Additional Tables 3. Wallis (1972) gives tables for regression models with quarterly data. Here you want to test for fourth-order autocorrelation rather than the first-order autocorrelation. In this case, the DW statistic is n d (û û t t 4 t= 5 4 = n 2 ûu t t=1 Wallis provides 5% critical values d L and d U for two situations: where the k regressors include an intercept (but not a full set of seasonal dummy variables) and another where the regressors include four quarterly seasonal dummy variables. In each case the critical values are for testing H 0 : ρ =0 against H 1 : ρ > 0. For the hypothesis H 1 : ρ < 0, Wallis suggests that the appropriate critical values are (4-d U ) and (4-d L ). King and Giles (1978) give further significance points for Wallis tests. 30 continued... ) 2

31 Additional Tables 4. King (1981) gives the 5% points for d L and d U quarterly time-series data with trend and/or seasonal dummy variables. These tables are for testing first-order autocorrelation. 5. King (1983) gives tables for the DW test for monthly data. In case of monthly data, you want to test for twelfth-order autocorrelation. 31

32 Nonparametric Runs Test (Gujarti, 1978) More general than the DW test. Interest in H 0 : ρ = 0. Test of AR(1) process in the error terms N+ = number of positive residuals N- = number of negative residuals N = number of observations Nr = number of runs Example: E + [ Nr ] = (2N N ) / N [ N ] = (2N N (2N N N)) /( N ( N 1)) VAR r Test Statistic: Z = (N r E(N r )) / VAR(N r ) N(0,1) Reject H 0 (non-autocorrelation) if the test statistic is too large in absolute value. 32

33 The REG Procedure Model: MODEL1 Dependent Variable: lnpcg Number of Observations Read 36 Number of Observations Used 36 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var continued...

34 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t

35 The REG Procedure Model: MODEL1 Dependent Variable: lnpcg Durbin-Watson D Pr < DW <.0001 Pr > DW Number of Observations 36 1st Order Autocorrelation NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. In this example, sample evidence exists to suggest the presence of positive serial correlation, which is the more common form of pattern in the residuals in regard to the use of economic or financial data. 35

36 Obs year lnpcgres Obs year lnpcgres

37 The Greene Problem In the Greene problem for gasoline, DW = and = Use of Nonparametric Runs test N = 36 N + = 19 N - = 17 N r = 11 E [ N ] VAR r = (2N + N ) / N = [ N ] = (2 N N (2 N N N )) /( N ( N 1)) = r ρˆ 37 Z = ( N E( N )) Z = = = at α =. 05, reject H :ρ = 0. Z crit r = 1.96 at α =.05 r VAR( N 0 r )

38 38 Analysis Limitations Analysts must recognize that a good Durbin-Watson statistic is insufficient evidence upon which to conclude that the error structure is contamination free in terms of autocorrelation. The Durbin-Watson test is only applicable for the presence of first-order autocorrelation. There is little reason to suppose that the correct model for residuals is AR(1). A mixed, autoregressive, movingaverage (ARMA) structure is much more likely to be correct, especially with quarterly, monthly, and weekly frequencies of time-series data. Modeling of the residuals can be employed following the methodology of Box and Jenkins (1976). Owing to higher frequencies of time-series data used in applied econometrics in recent years, the pattern of the error structure generally is more complex than the common AR(1) pattern.

39 A General Test for Higher-Order Serial Correlation The LM Test (Breusch and Pagan, 1980) LM - Lagrange multiplier y t = β u 0 t + β 1 = ρ 1 X u 1t t β + ρ u t 2 H0 : ρ1 = ρ2 =... = ρp = 2 k X 0 kt + u t ρ p u t t p = 1,2,..., n. + e t e t ~ IN(0, σ The Xs might or might not include lagged dependent variables. 1. Estimate by OLS and obtain the least squares residuals. ûγ t 2. Estimate. p u ˆ ˆ t = 0 + γ 1X1t γ k X kt + ut iρi + vt t= 1 3. Test whether the coefficients of û t i are all zero. Use the conventional F statistic.. 2 ) 39

40 40 Box-Pierce or Ljung-Box Tests Check the serial correlation pattern of the residuals. You must be sure that there is no serial correlation (desire white noise). H 0 : no pattern in the residuals (The residuals are white noise.) Box and Pierce (1970) suggest looking at not only the firstorder autocorrelation but autocorrelation of all orders of residuals. 2 r k Calculate Q = N m r 2 k, where is the autocorrelation of lag k, and N is the number of observations in the series. If the model fitted is appropriate,. Ljung and Box (1978) suggest a modification of the Q statistic for moderate sample sizes. Q* = N(N + 2) k= 1 m Q k= 1 (N 2 ~ & χ m k) 1 2 r k

41 Box-Pierce or Ljung-Box Tests With the Box-Pierce or Ljung-Box tests, you examine the interface of structural models with time-series models. Use the correlations and partial correlations of the residuals over time. The idea is to determine the appropriate pattern in the error structure from the autocorrelation and partial autocorrelation functions associated with the residuals. Autocorrelation functions tell you about moving average (MA) patterns. Partial autocorrelation functions tell you about autoregressive (AR) patterns. Anticipate ARMA error structures, particularly higherorder AR patterns in residuals of econometric models. 41

42 Section 6.4 Sample Problem: The Demand for Shrimp

43 The REG Procedure Model: MODEL1 Dependent Variable: QSHRIMP Number of Observations Read 97 Number of Observations Used 97 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var continued...

44 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept PSHRIMP <.0001 PFIN PSHELL ADSHRIMP ADFIN ADSHELL

45 The REG Procedure Output The REG Procedure Model: MODEL1 Dependent Variable: QSHRIMP Durbin-Watson D Pr < DW Pr > DW Number of Observations 97 1st Order Autocorrelation NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. Conclusion: No AR(1) pattern in the residuals 45

46 RESID

47 The ARIMA Procedure Output Name of Variable = resqshrimp Mean of Working Series -12E-16 Standard Deviation Number of Observations 97 The autocorrelation function (acf). A plot of the correlation of the residuals at various lags. Corr (et, et-k), k = 0, 1, 2,, 24. MA(3) Pattern Autocorrelations Lag Covariance Correlation Std Error ******************** * ** ****** ** * ** *** ** ***

48 Partial Autocorrelations Lag Correlation AR(3) Pattern * ** ****** ** The partial ** ** * ** * *** autocorrelation function (PACF). A plot of the correlation of the residuals at various lags after netting out intermittent lags **** ** ** * * ****. 48

49 PROC ARIMA Output Partial Autocorrelations Lag Correlation * *. Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations The Ljung Box Q* statistic reveals that the residual series is not white noise. 49

50 Correlogram of RESID Presence of MA(3), AR(3) pattern 50

51 Durbin-Watson Statistics Order DW Pr < DW Pr > DW NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. Godfrey's Serial Correlation Test Alternative LM Pr > LM 51 AR(1) AR(2) AR(3) Presence of AR(3) pattern

52 OLS Estimates Standard Approx Variable DF Estimate Error t Value Pr > t Intercept PSHRIMP <.0001 PFIN PSHELL ADSHRIMP ADFIN ADSHELL

53 Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(3,87) Obs*R-squared Prob. Chi-Square(3) Test Equation: Dependent Variable: RESID Method: Least Squares Sample: 1 97 Included observations: 97 Presample missing value lagged residuals set to zero. Coefficient Std. Error t-statistic Prob. C PSHRIMP PFIN PSHELL ADSHRIMP ADSHELL ADFIN RESID(-1) RESID(-2) RESID(-3) Presence of AR(3) Pattern 53 R-squared Mean dependent var -3.13E-15 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic)

54 Correcting for serial correlation through the use of PROC AUTOREG Use of Yule- Walker estimates of φ 1, φ 2, and φ 3 Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value Yule-Walker Estimates SSE DFE 87 MSE Root MSE SBC AIC Regress R-Square Total R-Square Log Likelihood Observations 97 54

55 Durbin-Watson Statistics Order DW Pr < DW Pr > DW NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. Now, no serial correlation exists in the residuals. 55

56 The AUTOREG Procedure Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) AR(2) AR(3) GLS Estimates Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 PSHRIMP <.0001 PFIN PSHELL ADSHRIMP ADFIN ADSHELL Statistically significant estimated coefficients for PSHRIMP and ADFIN

57 Partial Autocorrelations Preliminary MSE Starting values of estimates of φ 1, φ 2, and φ 3 in the ML procedure. Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value Algorithm converged. 57

58 Maximum Likelihood Estimates Use of the Maximum Likelihood procedure to produce estimates of φ 1, φ 2, and φ 3. SSE DFE 87 MSE Root MSE SBC AIC Regress R-Square Total R-Square Log Likelihood Observations 97 Durbin-Watson Statistics Order DW Pr < DW Pr > DW NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for use of autoreg procedure testing negative autocorrelation.

59 Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) AR(2) AR(3) ML Estimates Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 PSHRIMP <.0001 PFIN PSHELL ADSHRIMP ADFIN ADSHELL AR AR AR continued...

60 Autoregressive parameters assumed given. Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 PSHRIMP <.0001 PFIN PSHELL ADSHRIMP ADFIN ADSHELL

61 Depiction of the Estimated Model for the Qshrimp Problem The estimated model is based on the Maximum Likelihood estimates. Qshrimp t = *Pshrimp t *Pfin t *Pshell1 t *Adshrimp t Adfin t *Adshell1 t + v t 61 v t = *v t *v t *v t-3 + ε t MSE = (estimate of residual variance) This estimate is smaller than the OLS estimate of The total R-square statistic computed from the residuals of the autoregressive model is , reflecting the improved fit from the use of past residuals to help predict the next value of Qshrimp t. The Reg Rsq value is , which is the R-square statistic for a regression of transformed variables adjusted for the estimated autocorrelation.

62 Comparison of Diagnostic Statistics and Parameter Estimates from the Qshrimp Problem Explanatory Variables OLS Yule-Walker Maximum Likelihood Parameter Estimate Standard Error Parameter Estimate Standard Error Parameter Estimate Standard Error Intercept Pshrimp Pfin Pshell Adshrimp Adfin Adshell continued...

63 Comparison of Diagnostic Statistics and Parameter Estimates Parameter Estimate OLS Yule-Walker Maximum Likelihood Standard Error Parameter Estimate Standard Error Parameter Estimate MSE Root MSE SBC AIC Regress R- Square Total R-Square Log Likelihood AR 1 NA AR 2 NA AR 3 NA Standard Error 63 DW order DW order DW order

64 Section 6.5 Sample Problem: The Demand for Gasoline

65 Model: MODEL1 Dependent Variable: lnpcg Number of Observations Read 36 Number of Observations Used 36 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var continued...

66 OLS Estimates Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t

67 The REG Procedure Output Model: MODEL1 Dependent Variable: lnpcg Durbin-Watson D Pr < DW <.0001 Pr > DW Number of Observations 36 1st Order Autocorrelation NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. What is the conclusion regarding serial correlation based on the Durbin-Watson statistic? 67

68 RESID

69 MA(1) Name of Variable = lnpcgres Mean of Working Series 9.68E-16 Standard Deviation Number of Observations 36 Autocorrelations Lag Covariance Correlation Std Error ******************** ************ ************ * *** **** ** * E E **** ******* **** Statistically significant autocorrelation and partial autocorrelation coefficients 69

70 Partial Autocorrelations AR(1), AR(2) Lag Correlation ************ ********* ** ** ** ** ******* * ******. 70 continued...

71 Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations Statistically significant Ljung-Box Q* statistic. Thus, the pattern in the residuals is not random or white noise. 71

72 72

73 The AUTOREG Procedure Dependent Variable lnpcg Ordinary Least Squares Estimates SSE DFE 29 MSE Root MSE SBC AIC Regress R-Square Total R-Square Normal Test Pr > ChiSq Log Likelihood Observations 36 Durbin-Watson Statistics Order DW Pr < DW Pr > DW < NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. 73 continued...

74 AR(1), AR(2) Terms Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) AR(2) <.0001 Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t

75 The AUTOREG Procedure Estimates of Autocorrelations Lag Covariance Correlation ******************** ************ * Partial Autocorrelations Preliminary MSE Estimates of φ 1, φ 2 75 Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value continued...

76 Yule-Walker Estimates SSE DFE 27 MSE Root MSE SBC AIC Regress R-Square Total R-Square Log Likelihood Observations 36 Durbin-Watson Statistics Order DW Pr < DW Pr > DW After you take into account this pattern in the residuals, the serial correlation problem no longer is evident. NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. 76

77 The AUTOREG Procedure Output Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) AR(2) Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t

78 Estimates of Autocorrelations Lag Covariance Correlation ******************** ************ * Partial Autocorrelations Starting values in the ML procedure to obtain estimates of φ 1, φ 2 Preliminary MSE Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value continued...

79 Maximum Likelihood Estimates SSE DFE 27 MSE Root MSE SBC AIC Regress R-Square Total R-Square Log Likelihood Observations 36 Durbin-Watson Statistics Order DW Pr < DW Pr > DW NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. 79

80 Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) AR(2) ML Estimates Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t AR <.0001 AR continued...

81 Autoregressive parameters assumed given. Standard Approx Variable DF Estimate Error t Value Pr > t Intercept <.0001 lny <.0001 lnpg lnpnc lnpuc lnppt t

82 Depiction of the Estimated Model for the Demand for Gasoline Problem (Greene) The estimated model is based on the Maximum Likelihood estimates: ln pcg = *ln y *ln pg *ln *ln v t t puc t *ln t ppt = * vt * vt 2 t * t + ε t t pnc t Regress R-Square = Total R-Square = MSE = = DW order 1 = DW order 2 = σˆ 2 82

83 Section 6.6 A Test for Serial Correlation in the Presence of a Lagged Dependent Variable

84 Durbin s h-test A large sample test for autocorrelation when lagged dependent variables are present. H 0 :ρ = 0 AR(1) process in error terms ρˆ 1 (1/ 2)d h = ρˆ n /(1 nv( β)) d is the DW statistic. 84 The test breaks down if. Coefficient associated with ~ N(0,1) h & nv(ˆ) β 1 Yt 1 If the Durbin's h-test breaks down, compute the OLS residuals. Then regress û t on û t 1, yt 1, and the set of exogenous variables. The test û t 1 for ρ = 0 is carried out by testing the significance of the coefficient. (Durbin's m-test) û t

85 The REG Procedure Output Model: MODEL1 Dependent Variable: lnpcg Number of Observations Read 36 Number of Observations Used 35 Number of Observations with Missing Values 1 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var continued...

86 Parameter Estimates OLS Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept laglnpcg <.0001 lny lnpg <.0001 lnpnc lnpuc lnppt t Presence of a lagged dependent variable 86

87 The REG Procedure Output Model: MODEL1 Dependent Variable: lnpcg Durbin-Watson D Pr < DW Pr > DW Number of Observations 35 1st Order Autocorrelation NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation. The DW statistic reveals positive serial correlation, AR(1) pattern. 87

88 Name of Variable = lnpcgres Mean of Working Series -66E-17 Standard Deviation Number of Observations 35 Autocorrelations Lag Covariance Correlation Std Error ******************** **** ****** ** *** * **** * *

89 From the ACF and PACF plots, no pattern in the residuals is revealed. Partial Autocorrelations Lag Correlation **** ******* ****** *** ** *. The ARIMA Procedure Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations The Ljung-Box Q* statistic suggests a white noise residual series.

90 The AUTOREG Procedure Dependent Variable lnpcg Durbin s h-test, at least at the 0.10 level of significance, suggests a non-random residual series. Ordinary Least Squares Estimates SSE DFE 27 MSE Root MSE SBC AIC Regress R-Square Total R-Square Durbin h Pr > h Normal Test Pr > ChiSq Log Likelihood Observations 35 Durbin-Watson Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) continued...

91 Standard Approx Variable DF Estimate Error t Value Pr > t Intercept laglnpcg <.0001 lny lnpg <.0001 lnpnc lnpuc lnppt t

92 The AUTOREG Procedure Output Partial Autocorrelations Starting value of φ in the ML procedure Preliminary MSE Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value Algorithm converged.

93 Maximum Likelihood Estimates SSE DFE 26 MSE Root MSE SBC AIC Regress R-Square Total R-Square Log Likelihood Observations 35 Durbin-Watson Godfrey's Serial Correlation Test Alternative LM Pr > LM AR(1) GLS (ML) Estimates 93 continued...

94 Standard Approx Variable DF Estimate Error t Value Pr > t Intercept laglnpcg lny lnpg <.0001 lnpnc lnpuc lnppt t AR <

95 The AUTOREG Procedure Output Autoregressive parameters assumed given. Standard Approx Variable DF Estimate Error t Value Pr > t Intercept laglnpcg lny lnpg <.0001 lnpnc lnpuc lnppt t

96 Calculation of Durbin s h-test for the Greene Problem In the Greene Problem for gasoline demand: ˆ ρ = 1 (1.639 / 2) = n = 35 v( B) = ( ) 2 = h =

97 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > t Intercept Intercept lnpcgres laglnpcg lny lnpg lnpnc lnppt t

98 Section 6.7 Summary Remarks about the Issue of Serial Correlation

99 Final Considerations With time-series data, in most cases this problem will surface. Analysts must examine the error structure carefully. Minimally do the following: Graph the residuals over time. Consider the significance of the Durbin-Watson statistic. Consider higher-order autocorrelation structure via PROC ARIMA. Consider the Godfrey LM test. Consider the Box-Pierce or Ljung-Box tests (Q statistics). Re-estimate econometric models with AR(p) error structures via PROC AUTOREG. Use the Yule-Walker or Maximum Likelihood method to obtain estimates of the AR(P) error structure. A preference exists for the Maximum Likelihood method. 99