Section 2 NABE ASTEF 65

Transcription

1 Section 2 NABE ASTEF 65

2 Econometric (Structural) Models 66

3 67

4 The Multiple Regression Model 68

5 69

6 Assumptions 70

7 Components of Model Endogenous variables -- Dependent variables, values of which are determined within the system Exogenous variables -- Variables determined outside the system but influence the system by affecting the values of the endogenous variables Structural parameters -- Estimated using econometric techniques and relevant data Lagged endogenous variables Lagged exogenous variables Predetermined variables 71

8 72

9 Model Selection Criteria MSE T e 2 t t = = 1 T s 2 = T 2 et T t= 1 T = T K T T K MSE T is the " Penalty Factor" T K Akaike Information Criterion (AIC) 2K T AIC = e MSE 73

10 Schwarz or Bayesian Information Criterion (SIC) or (BIC) ( ) K T SIC BIC = T MSE 74

11 75

12 Elasticity: Interpretation: An elasticity of -.4 means a 1 percent change in the exogenous variable will lead to a -.4 percent change in the dependent variable. Suppose we wish to see whether or not the dependent variable and one independent variable are related after netting out the effect of any other independent variables in the model. Partial correlation coefficient between Y and X j -- the effect of X j on Y not accounted for by the other variables in the model. 76

13 77

14 Tests of Hypotheses About Single Coefficients k ) 78

15 Joint Tests on Several Regression Coefficients 79

16 80

17 Diagnostics Serial Correlation Collinearity Influence Points 81

18 Serial Correlation Definition Tests Durbin-Watson Test Nonparametric Runs Test Durbin h Test Lagrange Multiplier (LM) Test Box-Pierce Q Statistic Ljung- Box Q * Statistic (Small-sample modification of Box- Pierce Q Statistic) Generalized Least Squares 82

19 Serial Correlation Y i = β 0 + β1x1i + β E( ε ε ) 0 i j 2 X 2i β k X ki + ε i i = 1,2,..., n i j Disturbance terms are not independent. The correlation between εt and ε t k is called an autocorrelation of order k. 83

20 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. 84

21 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 may come from disciplines other than economics, finance, or business, for example, psychology and sociology. Or, these variables may represent factors that simply are difficult to quantify, such as tastes and preferences of consumers or technological innovation on the part of producers. 85

22 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. 86

23 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). 87

24 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; as well, the power of significance tests is reduced. With either positive or negative autocorrelation, least-squares parameter estimates usually are not as efficient as generalized least-squares parameter estimates. 88

25 Regression with Autocorrelated Errors 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, since the ordinary regression residuals are not independent, they contain information that can be used to improve the prediction of future values. 89

26 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 t = x β + ε t t ε t = φ ε 1 t 1 φ 2 ε t 2... φ m ε t m + v t v t ~ IN(0, σ 2 ) The notation indicates that each v t is normally and independently 2 distributed with mean 0 and variance σ 90

27 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 also is called the use of generalized least squares (GLS). 91

28 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. 92

29 Tests for Serial Correlation The Durbin-Watson Test H H 0 1 : ρ = 0 : ρ 0 0 DW = n t = 2 (ˆ e t t= 1 eˆ DW ( d) 2(1 ρ) n eˆ 2 t t 1 ) 2 4 ( or d statistic ) (approximation good only for large samples) If ρ = 0, then d = 2; if ρ = 1, then d = 0; if ρ = -1, then d = 4 93

30 The Distribution of DW (d) is reported by Durbin and Watson (1950, 1951) d L,d U depend on α, k, n DW invalid with models that contain no intercept and models that contain lagged dependent variables 94

31 The sampling distribution of d depends on the values of the exogenous variables and hence Durbin and Watson derived upper (du) limits and lower (dl) 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. 95

32 96

33 Appendix G, Statistical Table, cont. 97

34 Although the DW test is the most commonly used test for serial correlation, there are limitations: (1) tests for only first-order serial correlation. (2) test may be inconclusive. (3) test cannot be applied in models with lagged dependent variables. (4) test cannot be applied in models without intercepts. 98

35 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. 99

36 3. Wallis (1972) gives tables for regression models with quarterly data. Here one would like to test for fourth-order autocorrelation rather than first-order autocorrelation. In this case, the DW statistic is: n 2 (ût û t 4) t= 5 d4 = n 2 ût 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 these tests. 100

37 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, we may wish to test for twelfth-order autocorrelation. 101

38 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 Test Statistic: + [ Nr ] = (2N N ) / N [ N ] = (2N N (2N N N)) /( N ( N 1)) VAR r Z = (N r E(N r VAR(N N(0,1) Reject H 0 (non-autocorrelation) if the test statistic is too large in absolute value. )) / r ) 102

39 103

40 In this example, sample evidence exists to suggest the presence of positive serial correlation, the more common form of pattern in the residuals in regard to the use of economic or financial data. 104

41 105

42 In the Greene problem for gasoline, ρˆ DW=0.786 and rho= Use of Nonparametric Runs test N = 36 N + = 19 N - = 17 N r = 11 E [ N ] VAR r = (2N + N ) / N = [ ] N = (2N N (2N N N)) /( N ( N 1)) = r Z = ( N E( N )) Z = = = at α =. 05, reject H :ρ = 0. Z crit r = 1.96 at α =.05 r 0 VAR( N r ) 106

43 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. 107

44 Durbin h-test A large sample test for autocorrelation when lagged dependent variables are present. ρˆ 1 (1/ 2)d h = ρˆ n /(1 nv( β)) ~ h & Coefficient associated with N(0,1) Test breaks down if nv(ˆ) β 1 d is the DW statistic Yt 1 If the Durbin-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 for ρ = 0 is carried out by testing the significance of the coefficient û t 1 û t 108

45 OLS estimates Presence of a lagged dependent variable 109

46 Calculation of the Durbin 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 =

47 A General Test for higher-order Serial Correlation: The LM Test (Breusch and Pagan, 1980) LM - Lagrange multiplier y = β + β X β X + u t H 0 u : ρ 1 0 t = ρ 1 = ρ 2 1 u 1t t 1 + ρ 2 =... = ρ u p k t 2 = 0 kt t ρ p u t = 1,2,...,n. t p + e t e t ~ IN(0, σ The X s may or may not include lagged dependent variables. First: estimate by OLS and obtain the least squares residuals Second: estimate û t û t = γ0 + γ1x1t γ kxkt + û t iρi + p t= 1 v t. 2 ) û Third: test whether the coefficients of t i are all zero. Use the conventional F-statistic. 111

48 Box-Pierce or Ljung-Box Tests Check the serial correlation pattern of the residuals, need to be sure that there is no serial correlation (desire white noise) Box and Pierce (1970) suggest looking at not just the first-order autocorrelation but autocorrelation of all orders of residuals. 2 r k Calculate Q = N m 2 r k, where is the autocorrelation of lag k, and N is the number of observations in the series. If the model fitted is appropriate, number of estimated parameters k=1 Q 2 ~ & χ m p where p is the Ljung and Box (1978) suggest a modification of the Q-statistic for moderate sample sizes. Q* = N(N + 2) m k= 1 (N k) 1 2 r k 112

49 With the Box-Pierce of Ljung-Box tests, we examine the interface of structural models with timeseries models We 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 us about moving average (MA) patterns. Partial autocorrelation functions tell us about autoregressive (AR) patterns. Anticipate ARMA error structures, particularly higher-order AR patterns in residuals of econometric models. 113

50 114

51 The test can be used for different specifications of the error process: For Example: u ρ u e. t = 4 t 4 + t estimate u ˆ... ˆ t = γ + γ 1X1t + + γ k X kt + ρ4ut 4 + v 0 t. test H : ρ4 0 =

52 116

53 117

54 118

55 119

56 120

57 With time-series data, in most cases serial correlation problems will surface Analysts must examine the error structure carefully Minimally 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 121

58 Collinearity - Nature of Problem - Consequences - Introduction - Belsley, Kuh, Welsch Diagnostics Variance inflation factors Condition indices Variance-decomposition proportions - Circumvention of Problem: Ridge Regression 122

59 Collinearity Diagnostics Multicollinearity refers to the presence of highly intercorrelated explanatory variables in regression models. It is not surprising that it is considered one of the most ubiquitous, significant, and difficult problems in applied econometrics often referred to by modelers as the familiar curse. Collinearity diagnostics measure how closely regressors are related to other regressors and how this relationship affects the stability and variance of the regression estimates. 123

60 Signs of Multicollinearity Signs of multicollinearity in a regression analysis include: (1) Large standard errors on the regression coefficient, so that estimates of the true model parameters become unstable and low t-values prevail. (2) The parameter estimates vary considerably from sample to sample. (3) Often there will be drastic changes in the regression estimates after only minor data revision. (4) Conflicting conclusions will be reached from the usual tests of significance (such as the wrong sign for a parameter). 124

61 (5) Extreme correlations between pairs of variables. (6) Omitting a variable from the equation results in smaller regression standard errors. (7) A good fit not providing good forecasts. 125

62 We use multicollinearity diagnostics to: (1) produce a set of condition indices that signal the presence of one or more near dependencies among the variables. Linear dependency, an extreme form of multicollinearity, occurs when there is an exact linear relationship among the variables. (2) uncover those variables that are involved in particular near dependencies and to assess the degree to which the estimated regression coefficients are being degraded by the presence of the near dependencies. In practice, if one exogenous variable has a high squared multiple correlation (R-squared) with the other independent variables, it is extremely unlikely that the exogenous variable in question contributes significantly to the prediction equation. When the R-squared is too high, the variables is, in essence, redundant. 126

63 The Ballentine 127

64 Definition of Collinearity 128

65 Orthogonal Variables Non-orthogonal Variables Key Points: (1) Sampling variances of estimated OLS coefficients increase sharply. (2) Greater sampling covariances for the OLS coefficients. 129

66 Ill-Conditioning-Multicollinearity-Collinearity Deals with specific characteristics of the data matrix X a data problem, not a statistical problem Speak in terms of severity rather than of its existence or nonexistence Effects on structural integrity of econometric models Opposite of collinear orthogonal 130

67 Constitutes threat to proper specification and effective estimation of a structural relationship Larger variances (standard errors) of regression coefficients VAR not indistinguishable from the consequences of inadequate variability in regressors Covariance among parameter estimates often large and of the wrong sign Difficulties in interpretation Confidence regions for parameters wide Increase in type II error (Accept H0 when H0 false) Decrease power of tests 131

68 VIF i = 1 1 R 2 i 132

69 Condition Indices 133

70 134

71 the 135

72 136

73 137

74 three possible cases of degrading collinearity. 138

75 139

76 140

77 Ridge Regression 141

78 142

79 143

80 144

81 Influence Diagnostics and Robust Regression Concern Outliers Particular observations exert undue influence on regression results. Influential observations may be a legitimate part of data set. Influential observations may cast doubt either on the validity of the observation or on the general adequacy of the model. 145

82 Diagnostics 146

83 147

84 148

85 149

86 150

87 151

88 152

89 153

90 154

91 155

92 156

93 Influence on the Variance of Regression Coefficients 157

94 158

95 Robust Regression OLS suffers in performance in the presence of outliers and certain non-normal error distributions (heavy tails). Robust estimators are not sensitive to outliers. Robust estimators essentially weight down the influence of data points that produce residuals large in magnitude. 159

96 160

97 Cutoff DFFITS 2 p / n.89 DFBETAS 2 n.44 H 2p/n.40 COVRATIO 1+3p/n p/n

98 162

99 163

100 164

101 165