Chapter 5 Classical Linear Regression Model Assumptions and Diagnostics PDF
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Chris Brooks
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This document is a chapter on classical linear regression model assumptions and diagnostics. It details the assumptions, possible violations, and methods for detection and solutions in the context of econometrics and finance.
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Chapter 5 Classical linear regression model assumptions and diagnostics ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 1 Violation of the Assumptions of the CLRM Recall that we assumed of the CLRM disturbance terms: 1. E(ut) = 0...
Chapter 5 Classical linear regression model assumptions and diagnostics ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 1 Violation of the Assumptions of the CLRM Recall that we assumed of the CLRM disturbance terms: 1. E(ut) = 0 2. Var(ut) = 2 < 3. Cov (ui,uj) = 0 4. The X matrix is non-stochastic or fixed in repeated samples 5. ut N(0,2) ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 2 Investigating Violations of the Assumptions of the CLRM We will now study these assumptions further, and in particular look at: - How we test for violations - Causes - Consequences in general we could encounter any combination of 3 problems: - the coefficient estimates are wrong - the associated standard errors are wrong - the distribution that we assumed for the test statistics will be inappropriate - Solutions - the assumptions are no longer violated - we work around the problem so that we use alternative techniques which are still valid ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 3 Statistical Distributions for Diagnostic Tests Often, an F- and a 2- version of the test are available. The F-test version involves estimating a restricted and an unrestricted version of a test regression and comparing the RSS. The 2- version is sometimes called an “LM” test, and only has one degree of freedom parameter: the number of restrictions being tested, m. Asymptotically, the 2 tests are equivalent since the 2 is a special case of the F-distribution: 2 (m) → F (m, T − k ) as T − k → m For small samples, the F-version is preferable. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 4 Assumption 1: E(ut) = 0 Assumption that the mean of the disturbances is zero. For all diagnostic tests, we cannot observe the disturbances and so perform the tests of the residuals. The mean of the residuals will always be zero provided that there is a constant term in the regression. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 5 Assumption 2: Var(ut) = 2 < We have so far assumed that the variance of the errors is constant, 2 - this is known as homoscedasticity. û + If the errors do not have a t constant variance, we say that they are heteroscedastic e.g. say we estimate a regression and calculate the residuals, ut. x 2t - ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 6 Detection of Heteroscedasticity: The GQ Test Graphical methods Formal tests: There are many of them: we will discuss Goldfeld-Quandt test and White’s test The Goldfeld-Quandt (GQ) test is carried out as follows. 1. Split the total sample of length T into two sub-samples of length T1 and T2. The regression model is estimated on each sub-sample and the two residual variances are calculated. 2. The null hypothesis is that the variances of the disturbances are equal, H0 : 1 = 2 2 2 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 7 The GQ Test (Cont’d) 3. The test statistic, denoted GQ, is simply the ratio of the two residual variances where the larger of the two variances must be placed in the numerator. s12 GQ = 2 s2 4. The test statistic is distributed as an F(T1-k, T2-k) under the null of homoscedasticity. 5. A problem with the test is that the choice of where to split the sample is that usually arbitrary and may crucially affect the outcome of the test. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 8 Detection of Heteroscedasticity using White’s Test White’s general test for heteroscedasticity is one of the best approaches because it makes few assumptions about the form of the heteroscedasticity. The test is carried out as follows: 1. Assume that the regression we carried out is as follows yt = 1 + 2x2t + 3x3t + ut And we want to test Var(ut) = 2. We estimate the model, obtaining the residuals, ut 2. Then run the auxiliary regression uˆt2 = 1 + 2 x2t + 3 x3t + 4 x22t + 5 x32t + 6 x2t x3t + vt ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 9 Performing White’s Test for Heteroscedasticity 3. Obtain R2 from the auxiliary regression and multiply it by the number of observations, T. It can be shown that T R2 2 (m) where m is the number of regressors in the auxiliary regression excluding the constant term. 4. If the 2 test statistic from step 3 is greater than the corresponding value from the statistical table then reject the null hypothesis that the disturbances are homoscedastic. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 10 Consequences of Using OLS in the Presence of Heteroscedasticity OLS estimation still gives unbiased coefficient estimates, but they are no longer BLUE. This implies that if we still use OLS in the presence of heteroscedasticity, our standard errors could be inappropriate and hence any inferences we make could be misleading. Whether the standard errors calculated using the usual formulae are too big or too small will depend upon the form of the heteroscedasticity. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 11 How Do we Deal with Heteroscedasticity? If the form (i.e. the cause) of the heteroscedasticity is known, then we can use an estimation method which takes this into account (called generalised least squares, GLS). A simple illustration of GLS is as follows: Suppose that the error variance is related to another variable zt by var (ut ) = 2 zt2 To remove the heteroscedasticity, divide the regression equation by zt yt 1 x x = 1 + 2 2t + 3 3t + vt zt zt zt zt ut where vt = is an error term. zt u var (ut ) 2 zt2 Now var (vt ) = var t = 2 = 2 = 2 for known zt. zt z t z t ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 12 Other Approaches to Dealing with Heteroscedasticity So the disturbances from the new regression equation will be homoscedastic. Other solutions include: 1. Transforming the variables into logs or reducing by some other measure of “size”. 2. Use White’s heteroscedasticity consistent standard error estimates. The effect of using White’s correction is that in general the standard errors for the slope coefficients are increased relative to the usual OLS standard errors. This makes us more “conservative” in hypothesis testing, so that we would need more evidence against the null hypothesis before we would reject it. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 13 Background – The Concept of a Lagged Value t yt yt-1 yt 1989M09 0.8 - - 1989M10 1.3 0.8 1.3-0.8=0.5 1989M11 -0.9 1.3 -0.9-1.3=-2.2 1989M12 0.2 -0.9 0.2--0.9=1.1 1990M01 -1.7 0.2 -1.7-0.2=-1.9 1990M02 2.3 -1.7 2.3--1.7=4.0 1990M03 0.1 2.3 0.1-2.3=-2.2 1990M04 0.0 0.1 0.0-0.1=-0.1............ ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 14 Autocorrelation We assumed of the CLRM’s errors that Cov (ui , uj) = 0 for ij, i.e. This is essentially the same as saying there is no pattern in the errors. Obviously we never have the actual u’s, so we use their sample counterpart, the residuals (the ut’s). If there are patterns in the residuals from a model, we say that they are autocorrelated. Some stereotypical patterns we may find in the residuals are given on the next 3 slides. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 15 Positive Autocorrelation + û t û t + - + uˆ t −1 Time - - Positive Autocorrelation is indicated by a cyclical residual plot over time. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 16 Negative Autocorrelation + û t û t + - uˆ+t −1 Time - - Negative autocorrelation is indicated by an alternating pattern where the residuals cross the time axis more frequently than if they were distributed randomly ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 17 No pattern in residuals – No autocorrelation û t + + û t - + uˆ t −1 - - No pattern in residuals at all: this is what we would like to see ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 18 Detecting Autocorrelation: The Durbin-Watson Test The Durbin-Watson (DW) is a test for first order autocorrelation - i.e. it assumes that the relationship is between an error and the previous one ut = ut-1 + vt (1) where vt N(0, v2). The DW test statistic actually tests H0 : =0 and H1 : 0 The test statistic is calculated by T ( ut − ut −1) 2 DW = t = 2 T ut 2 t =2 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 19 The Durbin-Watson Test: Critical Values We can also write DW 2(1 − ) (2) where is the estimated correlation coefficient. Since is a correlation, it implies that − 1 pˆ 1. Rearranging for DW from (2) would give 0DW4. If = 0, DW = 2. So roughly speaking, do not reject the null hypothesis if DW is near 2 → i.e. there is little evidence of autocorrelation Unfortunately, DW has 2 critical values, an upper critical value (du) and a lower critical value (dL), and there is also an intermediate region where we can neither reject nor not reject H0. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 20 The Durbin-Watson Test: Interpreting the Results Conditions which Must be Fulfilled for DW to be a Valid Test 1. Constant term in regression 2. Regressors are non-stochastic 3. No lags of dependent variable ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 21 Another Test for Autocorrelation: The Breusch-Godfrey Test It is a more general test for rth order autocorrelation: ut = 1ut −1 + 2ut − 2 + 3ut − 3 +...+ r ut − r + vt , vt N(0, v2 ) The null and alternative hypotheses are: H0 : 1 = 0 and 2 = 0 and... and r = 0 H1 : 1 0 or 2 0 or... or r 0 The test is carried out as follows: 1. Estimate the linear regression using OLS and obtain the residuals, ut. 2. Regress ut on all of the regressors from stage 1 (the x’s) plus ut −1 , ut − 2 ,..., ut − r Obtain R2 from this regression. 3. It can be shown that (T-r)R2 2(r) If the test statistic exceeds the critical value from the statistical tables, reject the null hypothesis of no autocorrelation. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 22 Consequences of Ignoring Autocorrelation if it is Present The coefficient estimates derived using OLS are still unbiased, but they are inefficient, i.e. they are not BLUE, even in large sample sizes. Thus, if the standard error estimates are inappropriate, there exists the possibility that we could make the wrong inferences. R2 is likely to be inflated relative to its “correct” value for positively correlated residuals. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 23 “Remedies” for Autocorrelation If the form of the autocorrelation is known, we could use a GLS procedure – i.e. an approach that allows for autocorrelated residuals e.g., Cochrane-Orcutt. But such procedures that “correct” for autocorrelation require assumptions about the form of the autocorrelation. If these assumptions are invalid, the cure would be more dangerous than the disease! - see Hendry and Mizon (1978). However, it is unlikely to be the case that the form of the autocorrelation is known, and a more “modern” view is that residual autocorrelation presents an opportunity to modify the regression. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 24 Dynamic Models All of the models we have considered so far have been static, e.g. yt = 1 + 2x2t +... + kxkt + ut But we can easily extend this analysis to the case where the current value of yt depends on previous values of y or one of the x’s, e.g. yt = 1 + 2x2t +... + kxkt + 1yt-1 + 2x2t-1 + … + kxkt-1+ ut We could extend the model even further by adding extra lags, e.g. x2t-2 , yt-3. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 25 Why Might we Want/Need To Include Lags in a Regression? Inertia of the dependent variable Over-reactions Measuring time series as overlapping moving averages However, other problems with the regression could cause the null hypothesis of no autocorrelation to be rejected: – Omission of relevant variables, which are themselves autocorrelated. – If we have committed a “misspecification” error by using an inappropriate functional form. – Autocorrelation resulting from unparameterised seasonality. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 26 Models in First Difference Form Another way to sometimes deal with the problem of autocorrelation is to switch to a model in first differences. Denote the first difference of yt, i.e. yt - yt-1 as yt; similarly for the x- variables, x2t = x2t - x2t-1 etc. The model would now be yt = 1 + 2 x2t +... + kxkt + ut Sometimes the change in y is purported to depend on previous values of y or xt as well as changes in x: yt = 1 + 2 x2t + 3x2t-1 +4yt-1 + ut ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 27 The Long Run Static Equilibrium Solution One interesting property of a dynamic model is its long run or static equilibrium solution. “Equilibrium” implies that the variables have reached some steady state and are no longer changing, i.e. if y and x are in equilibrium, we can say yt = yt+1 =... =y and xt = xt+1 =... =x Consequently, yt = yt - yt-1 = y - y = 0 etc. So the way to obtain a long run static solution is: 1. Remove all time subscripts from variables 2. Set error terms equal to their expected values, E(ut)=0 3. Remove first difference terms altogether 4. Gather terms in x together and gather terms in y together. These steps can be undertaken in any order ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 28 The Long Run Static Equilibrium Solution: An Example If our model is yt = 1 + 2 x2t + 3x2t-1 +4yt-1 + ut then the static solution would be given by 0 = 1 + 3x2t-1 +4yt-1 4yt-1 = - 1 - 3x2t-1 − 1 3 y= − x2 4 4 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 29 Problems with Adding Lagged Regressors to “Cure” Autocorrelation Inclusion of lagged values of the dependent variable violates the assumption that the RHS variables are non-stochastic. What does an equation with a large number of lags actually mean? Note that if there is still autocorrelation in the residuals of a model including lags, then the OLS estimators will not even be consistent. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 30 Multicollinearity This problem occurs when the explanatory variables are very highly correlated with each other. Perfect multicollinearity Cannot estimate all the coefficients - e.g. suppose x3 = 2x2 and the model is yt = 1 + 2x2t + 3x3t + 4x4t + ut Problems if Near Multicollinearity is Present but Ignored - R2 will be high but the individual coefficients will have high standard errors. - The regression becomes very sensitive to small changes in the specification. - Thus confidence intervals for the parameters will be very wide, and significance tests might therefore give inappropriate conclusions. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 31 Measuring Multicollinearity The easiest way to measure the extent of multicollinearity is simply to look at the matrix of correlations between the individual variables. e.g. Corr x2 x3 x4 x2 - 0.2 0.8 x3 0.2 - 0.3 x4 0.8 0.3 - But another problem: if 3 or more variables are linear - e.g. x2t + x3t = x4t Note that high correlation between y and one of the x’s is not muticollinearity. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 32 Solutions to the Problem of Multicollinearity “Traditional” approaches, such as ridge regression or principal components. But these usually bring more problems than they solve. Some econometricians argue that if the model is otherwise OK, just ignore it The easiest ways to “cure” the problems are - drop one of the collinear variables - transform the highly correlated variables into a ratio - go out and collect more data e.g. - a longer run of data - switch to a higher frequency ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 33 Adopting the Wrong Functional Form We have previously assumed that the appropriate functional form is linear. This may not always be true. We can formally test this using Ramsey’s RESET test, which is a general test for mis-specification of functional form. Essentially the method works by adding higher order terms of the fitted values (e.g. yt2 , yt3 etc.) into an auxiliary regression: Regress ut on powers of the fitted values: ut = 0 + 1 yt2 + 2 yt3 +...+ p −1 ytp + vt Obtain R2 from this regression. The test statistic is given by TR2 and is distributed as a 2 ( p − 1). So if the value of the test statistic is greater than a 2 ( p − 1) then reject the null hypothesis that the functional form was correct. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 34 But what do we do if this is the case? The RESET test gives us no guide as to what a better specification might be. One possible cause of rejection of the test is if the true model is yt = 1 + 2 x2t + 3 x22t + 4 x23t + ut In this case the remedy is obvious. Another possibility is to transform the data into logarithms. This will linearise many previously multiplicative models into additive ones: yt = Axt eut ln yt = + ln xt + ut ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 35 Testing the Normality Assumption Why did we need to assume normality for hypothesis testing? Testing for Departures from Normality The Bera Jarque normality test A normal distribution is not skewed and is defined to have a coefficient of kurtosis of 3. The kurtosis of the normal distribution is 3 so its excess kurtosis (b2-3) is zero. Skewness and kurtosis are the (standardised) third and fourth moments of a distribution. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 36 Normal versus Skewed Distributions f(x) f(x) x x A normal distribution A skewed distribution ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 37 Leptokurtic versus Normal Distribution 0.5 0.4 0.3 0.2 0.1 0.0 -5.4 -3.6 -1.8 -0.0 1.8 3.6 5.4 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 38 Testing for Normality Bera and Jarque formalise this by testing the residuals for normality by testing whether the coefficient of skewness and the coefficient of excess kurtosis are jointly zero. It can be proved that the coefficients of skewness and kurtosis can be expressed respectively as: E[u3 ] E[u4 ] b1 = and b2 = 2 2 2 3/ 2 ( ) ( ) The Bera Jarque test statistic is given by b12 (b2 − 3)2 W =T + ~ 2 (2) 6 24 We estimate b1 and b2 using the residuals from the OLS regression, u. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 39 What do we do if we find evidence of Non-Normality? It is not obvious what we should do! Could use a method which does not assume normality, but difficult and what are its properties? Often the case that one or two very extreme residuals causes us to reject the normality assumption. An alternative is to use dummy variables. e.g. say we estimate a monthly model of asset returns from 1980-1990, and we plot the residuals, and find a particularly large outlier for October 1987: ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 40 What do we do if we find evidence of Non-Normality? (cont’d) û t + Oct Time 1987 - Create a new variable: D87M10t = 1 during October 1987 and zero otherwise. This effectively knocks out that observation. But we need a theoretical reason for adding dummy variables. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 41 Omission of an Important Variable or Inclusion of an Irrelevant Variable Omission of an Important Variable Consequence: The estimated coefficients on all the other variables will be biased and inconsistent unless the excluded variable is uncorrelated with all the included variables. Even if this condition is satisfied, the estimate of the coefficient on the constant term will be biased. The standard errors will also be biased. Inclusion of an Irrelevant Variable Coefficient estimates will still be consistent and unbiased, but the estimators will be inefficient. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 42 Parameter Stability Tests So far, we have estimated regressions such as yt = 1 + 2x2t + 3x3t + ut We have implicitly assumed that the parameters (1, 2 and 3) are constant for the entire sample period. We can test this implicit assumption using parameter stability tests. The idea is essentially to split the data into sub-periods and then to estimate up to three models, for each of the sub-parts and for all the data and then to “compare” the RSS of the models. There are two types of test we can look at: - Chow test (analysis of variance test) - Predictive failure tests ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 43 The Chow Test The steps involved are: 1. Split the data into two sub-periods. Estimate the regression over the whole period and then for the two sub-periods separately (3 regressions). Obtain the RSS for each regression. 2. The restricted regression is now the regression for the whole period while the “unrestricted regression” comes in two parts: for each of the sub- samples. We can thus form an F-test which is the difference between the RSS’s. RSS − ( RSS1 + RSS2 ) T − 2k The statistic is RSS1 + RSS2 k ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 44 The Chow Test (cont’d) where: RSS = RSS for whole sample RSS1 = RSS for sub-sample 1 RSS2 = RSS for sub-sample 2 T = number of observations 2k = number of regressors in the “unrestricted” regression (since it comes in two parts) k = number of regressors in (each part of the) “unrestricted” regression 3. Perform the test. If the value of the test statistic is greater than the critical value from the F-distribution, which is an F(k, T-2k), then reject the null hypothesis that the parameters are stable over time. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 45 A Chow Test Example Consider the following regression for the CAPM (again) for the returns on Glaxo. Say that we are interested in estimating Beta for monthly data from 1981-1992. The model for each sub-period is 1981M1 - 1987M10 0.24 + 1.2RMt T = 82 RSS1 = 0.03555 1987M11 - 1992M12 0.68 + 1.53RMt T = 62 RSS2 = 0.00336 1981M1 - 1992M12 0.39 + 1.37RMt T = 144 RSS = 0.0434 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 46 A Chow Test Example - Results The null hypothesis is H0: 1 = 2 and 1 = 2 The unrestricted model is the model where this restriction is not imposed 00434. − ( 00355. + 000336. ) 144 − 4 Test statistic = 00355. + 000336. 2 = 7.698 Compare with 5% F(2,140) = 3.06 We reject H0 at the 5% level and say that we reject the restriction that the coefficients are the same in the two periods. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 47 The Predictive Failure Test Problem with the Chow test is that we need to have enough data to do the regression on both sub-samples, i.e. T1>>k, T2>>k. An alternative formulation is the predictive failure test. What we do with the predictive failure test is estimate the regression over a “long” sub-period (i.e. most of the data) and then we predict values for the other period and compare the two. To calculate the test: - Run the regression for the whole period (the restricted regression) and obtain the RSS - Run the regression for the “large” sub-period and obtain the RSS (called RSS1). Note we call the number of observations T1 (even though it may come second). RSS − RSS1 T1 − k Test Statistic = RSS1 T2 where T2 = number of observations we are attempting to “predict”. The test statistic will follow an F(T2, T1-k). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 48 Backwards versus Forwards Predictive Failure Tests There are 2 types of predictive failure tests: - Forward predictive failure tests, where we keep the last few observations back for forecast testing, e.g. we have observations for 1970Q1-1994Q4. So estimate the model over 1970Q1-1993Q4 and forecast 1994Q1-1994Q4. - Backward predictive failure tests, where we attempt to “back-cast” the first few observations, e.g. if we have data for 1970Q1-1994Q4, and we estimate the model over 1971Q1-1994Q4 and backcast 1970Q1-1970Q4. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 49 Predictive Failure Tests – An Example We have the following models estimated: For the CAPM on Glaxo. 1980M1-1991M12 0.39 + 1.37RMt T = 144 RSS = 0.0434 1980M1-1989M12 0.32 + 1.31RMt T1 = 120 RSS1 = 0.0420 Can this regression adequately “forecast” the values for the last two years? 0.0434 − 0.0420 120 − 2 Test Statistic = = 0.164 0.0420 24 Compare with F(24,118) = 1.66. So we do not reject the null hypothesis that the model can adequately predict the last few observations. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 50 How do we decide the sub-parts to use? As a rule of thumb, we could use all or some of the following: - Plot the dependent variable over time and split the data accordingly to any obvious structural changes in the series, e.g. 1400 1200 1000 Value of Series (yt) 800 600 400 200 0 105 131 157 183 209 235 261 287 313 339 365 391 417 443 27 53 79 1 Sample Period - Split the data according to any known important historical events (e.g. stock market crash, new government elected) - Use all but the last few observations and do a predictive failure test on those. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 51 Measurement Errors If there is measurement error in one or more of the explanatory variables, this will violate the assumption that the explanatory variables are non- stochastic Sometimes this is also known as the errors-in-variables problem Measurement errors can occur in a variety of circumstances, e.g. – Macroeconomic variables are almost always estimated quantities (GDP, inflation, and so on), as is most information contained in company accounts – Sometimes we cannot observe or obtain data on a variable we require and so we need to use a proxy variable – for instance, many models include expected quantities (e.g., expected inflation) but we cannot typically measure expectations. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 52 Measurement Error in the Explanatory Variable(s) Suppose that we wish to estimate a model containing just one explanatory variable, xt: yt = β1 + β2xt + ut, where ut is a disturbance term Suppose further that xt is measured with error so that instead of observing its true value, we observe a noisy version, , that comprises the actual xt plus some additional noise, vt that is independent of xt and ut: Taking the first equation and substituting in for xt from the second: We can rewrite this equation by separately expressing the composite error term, (ut − β2vt) ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 53 Measurement Error in the Explanatory Variable(s) It should be clear from this equation and the one for the explanatory variable measured with error, and the composite error term, (ut − β2vt), are correlated since both depend on vt Thus the requirement that the explanatory variables are non-stochastic does not hold This causes the parameters to be estimated inconsistently The size of the bias in the estimates will be a function of the variance of the noise in xt as a proportion of the overall disturbance variance If β2 is positive, the bias will be negative but if β2 is negative, the bias will be positive So the parameter estimate will always be biased towards zero as a result of the measurement noise. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 54 Measurement Error and Tests of the CAPM The standard approach to testing the CAPM pioneered by Fama and MacBeth (1973) comprises two stages Since the betas are estimated at the first stage rather than being directly observable, they will surely contain measurement error The effect of this has sometimes been termed attenuation bias. Tests of the CAPM showed that the relationship between beta and returns was smaller than expected, and this is precisely what would happen as a result of measurement error Various approaches to solving this issue have been proposed, the most common of which is to use portfolio betas in place of individual betas An alternative approach (Shanken,1992) is to modify the standard errors in the second stage regression to adjust directly for the measurement errors. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 55 Measurement Error in the Explained Variable Measurement error in the explained variable is much less serious than in the explanatory variable(s) This is one of the motivations for the inclusion of the disturbance term in a regression model When the explained variable is measured with error, the disturbance term will in effect be a composite of the usual disturbance term and another source of noise from the measurement error Then the parameter estimates will still be consistent and unbiased and the usual formulae for calculating standard errors will still be appropriate The only consequence is that the additional noise means the standard errors will be enlarged relative to the situation where there was no measurement error in y. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 56 A Strategy for Building Econometric Models Our Objective: To build a statistically adequate empirical model which - satisfies the assumptions of the CLRM - is parsimonious - has the appropriate theoretical interpretation - has the right “shape” - i.e. - all signs on coefficients are “correct” - all sizes of coefficients are “correct” - is capable of explaining the results of all competing models ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 57 2 Approaches to Building Econometric Models There are 2 popular philosophies of building econometric models: the “specific-to-general” and “general-to-specific” approaches. “Specific-to-general” was used almost universally until the mid 1980’s, and involved starting with the simplest model and gradually adding to it. Little, if any, diagnostic testing was undertaken. But this meant that all inferences were potentially invalid. An alternative and more modern approach to model building is the “LSE” or Hendry “general-to-specific” methodology. The advantages of this approach are that it is statistically sensible and also the theory on which the models are based usually has nothing to say about the lag structure of a model. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 58 The General-to-Specific Approach First step is to form a “large” model with lots of variables on the right hand side This is known as a GUM (generalised unrestricted model) At this stage, we want to make sure that the model satisfies all of the assumptions of the CLRM If the assumptions are violated, we need to take appropriate actions to remedy this, e.g. - taking logs - adding lags - dummy variables We need to do this before testing hypotheses Once we have a model which satisfies the assumptions, it could be very big with lots of lags & independent variables ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 59 The General-to-Specific Approach: Reparameterising the Model The next stage is to reparameterise the model by - knocking out very insignificant regressors - some coefficients may be insignificantly different from each other, so we can combine them. At each stage, we need to check the assumptions are still OK. Hopefully at this stage, we have a statistically adequate empirical model which we can use for - testing underlying financial theories - forecasting future values of the dependent variable - formulating policies, etc. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 60 Regression Analysis In Practice - A Further Example: Determinants of Sovereign Credit Ratings Cantor and Packer (1996) Financial background: What are sovereign credit ratings and why are we interested in them? Two ratings agencies (Moody’s and Standard and Poor’s) provide credit ratings for many governments. Each possible rating is denoted by a grading: Moody’s Standard and Poor’s Aaa AAA …… ….. B3 B- ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 61 Purposes of the Paper - to attempt to explain and model how the ratings agencies arrived at their ratings. - to use the same factors to explain the spreads of sovereign yields above a risk-free proxy - to determine what factors affect how the sovereign yields react to ratings announcements ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 62 Determinants of Sovereign Ratings Data Quantifying the ratings (dependent variable): Aaa/AAA=16,... , B3/B-=1 Explanatory variables (units of measurement): - Per capita income in 1994 (thousands of dollars) - Average annual GDP growth 1991-1994 (%) - Average annual inflation 1992-1994 (%) - Fiscal balance: Average annual government budget surplus as a proportion of GDP 1992-1994 (%) - External balance: Average annual current account surplus as a proportion of GDP 1992-1994 (%) - External debt Foreign currency debt as a proportion of exports 1994 (%) - Dummy for economic development - Dummy for default history Income and inflation are transformed to their logarithms. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 63 The model: Linear and estimated using OLS Dependent Variable Expected Average Moody’s S&P Moody’s / S&P Explanatory Variable sign Rating Rating Rating Difference Intercept ? 1.442 3.408 -0.524 3.932** (0.663) (1.379) (-0.223) (2.521) Per capita income + 1.242*** 1.027*** 1.458*** -0.431*** (5.302) (4.041) (6.048) (-2.688) GDP growth + 0.151 0.130 0.171** -0.040 (1.935) (1.545) (2.132) (0.756) Inflation - -0.611*** -0.630*** -0.591*** -0.039 (-2.839) (-2.701) (2.671) (-0.265) Fiscal Balance + 0.073 0.049 0.097* -0.048 (1.324) (0.818) (1.71) (-1.274) External Balance + 0.003 0.006 0.001 0.006 (0.314) (0.535) (0.046) (0.779) External Debt - -0.013*** -0.015*** -0.011*** -0.004*** (-5.088) (-5.365) (-4.236) (-2.133) Development dummy + 2.776*** 2.957*** 2.595*** 0.362 (4.25) (4.175) (3.861) (0.81) Default dummy - -2.042*** -1.63** -2.622*** 1.159*** (-3.175) (-2.097) (-3.962) (2.632) Adjusted R2 0.924 0.905 0.926 0.836 Notes: t-ratios in parentheses; *, **, and *** indicate significance at the 10%, 5% and 1% levels respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 64 Interpreting the Model From a statistical perspective Virtually no diagnostics Adjusted R2 is high Look at the residuals: actual rating - fitted rating From a financial perspective Do the coefficients have their expected signs and sizes? Do Ratings Add to Publicly Available Available Information? Now dependent variable is - Log (Yield on the sovereign bond - yield on a US treasury bond) ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 65 Do Ratings Add to Publicly Available Available Information? Results Dependent Variable: Log (yield spread) Variable Expected Sign (1) (2) (3) Intercept ? 2.105*** 0.466 0.074 (16.148) (0.345) (0.071) Average - -0.221*** -0.218*** Rating (-19.175) (-4.276) Per capita - -0.144 0.226 income (-0.927) (1.523) GDP growth - -0.004 0.029 (-0.142) (1.227) Inflation + 0.108 -0.004 (1.393) (-0.068) Fiscal Balance - -0.037 -0.02 (-1.557) (-1.045) External - -0.038 -0.023 Balance (-1.29) (-1.008) External Debt + 0.003*** 0.000 (2.651) (0.095) Development - -0.723*** -0.38 dummy (-2.059) (-1.341) Default dummy + 0.612*** 0.085 (2.577) (0.385) Adjusted R2 0.919 0.857 0.914 Notes: t-ratios in parentheses; *, **, and *** indicate significance at the 10%, 5% and 1% levels respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 66 What Determines How the Market Reacts to Ratings Announcements? The sample: Every announcement of a ratings change that occurred between 1987 and 1994 - 79 such announcements spread over 18 countries. 39 were actual ratings changes 40 were “watchlist / outlook” changes The dependent variable: changes in the relative spreads over the US T-bond over a 2-day period at the time of the announcement. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 67 What Determines How the Market Reacts to Ratings Announcements? Explanatory variables. 0 /1 dummies for - Whether the announcement was positive - Whether there was an actual ratings change - Whether the bond was speculative grade - Whether there had been another ratings announcement in the previous 60 days. and - The change in the spread over the previous 60 days. - The ratings gap between the announcing and the other agency ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 68 What Determines How the Market Reacts to Ratings Announcements? Results Dependent Variable: Log Relative Spread Independent variable Coefficient (t-ratio) Intercept -0.02 (-1.4) Positive announcements 0.01 (0.34) Ratings changes -0.01 (-0.37) Moody’s announcements 0.02 (1.51) Speculative grade 0.03** (2.33) Change in relative spreads from day –60 to day -1 -0.06 (-1.1) Rating gap 0.03* (1.7) Other rating announcements from day –60 to day -1 0.05** (2.15) 2 Adjusted R 0.12 Note: * and ** denote significance at the 10% and 5% levels respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 69 Conclusions 6 factors appear to play a big role in determining sovereign credit ratings - incomes, GDP growth, inflation, external debt, industrialised or not, and default history. The ratings provide more information on yields than all of the macro factors put together. We cannot determine well what factors influence how the markets will react to ratings announcements. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 70 Comments on the Paper Only 49 observations for first set of regressions and 35 for yield regressions and up to 10 regressors No attempt at reparameterisation Little attempt at diagnostic checking Where did the factors (explanatory variables) come from? ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 71
