Normality assumption of the classical linear regression model

Normality assumption of the classical linear regression model One of the assumptions of the CLRM is that the residuals ...





Normality assumption of the classical linear regression model




One of the assumptions of the CLRM is that the residuals are normally distributed with zero mean and variance. The normality assumption is one of the most misunderstood in all statistics(get more details here). The violation of this assumption leads to invalid inferential statistics of the model. There are various test for normality e.g Histogram of the residuals, Normal probability plots, Jarque-Bera test e.t.c(Gujarati, 2004).
We will consider the Jarque–Bera (JB) test. The test is still quite popular in practice because the first four moments are often of particular interest, and deviations from the normal distribution beyond that may not be of equal importance(Lutkepohl, 2004).




Jarque–Bera (JB) Test



The JB test of normality is an asymptotic, or large-sample, test. It is also based on the OLS residuals.
Normality test was introduced by Fisher (1948), but, a standard Jarque-Bera (1980) test is being used widely. It is expressed in terms of the third and fourth moments of the disturbances, as follows:
The null hypothesis is H0 and the residuals are normally distributed


Residuals: Ût = Yt – Ȃ1X1t – Ȃ2X2t  – … – ȂkXkt    [1]


 µ2=   ΣÛ2/n, µ3= ΣÛ3/n, µ4= ΣÛ4/n


Jarque- Bera Statistic


JB= [n-k] [(s2/6) + ((k-3)2/24)]             [2]


Where:


S: measure skewness coefficient, K: measure Kurtosis coefficient


Note that the µ3 equals skewness of the residuals while µ4 is the kurtosis of the residuals


The values for K and S when the variable is normally distributed:

K=3 and S=0. The JB statistic is expected to be zero.

The errors are normal if they are not skewed (S=0) and not leptokurtic (K=3). Normality means the S should be close to zero and K should be close to 3 and the errors will be insignificant.

Jarque and Bera showed that in large sample the JB statistic given above follows the chi-square distribution with 2 degree of freedom. If the p value of the JB statistic is low, especially if the value of the statistic is different from 0, one can reject the hypothesis that the errors are normally distributed. But if the p value is satisfactorily high, especially if the value of the statistic is close to zero, we do not reject the normality assumption. Hence, strictly speaking one should not use the JB statistic in a small sample(Gujarati, 2004).

Computer Application(Eviews)


We need to check on the histogram and the j-B statistic in order to check for normality of residuals in a regression model. To do this we need to estimate the equation using the data from previous post(here). We can estimate the equation either by typing in the command line or by choosing QUICK>>ESTIMATE EQUATION, then specify the equation (using ‘‘Y c X’’ with sample from 1970 to 2017) and click <OK>.
To check for normality, click from the result output VIEW>>RESIDUAL DIAGNOSTICS>>HISTOGRAM-NORMALITY TEST.



From the histogram graph, we see that the residuals are not normally distributed. At the right hand corner we can see the J-B statistic, skewness, kurtosis and probability value. The J-B statistics is about 10.69020 , the residuals are negatively skewed (-0.124498) and kurtosis(2.380786) which means its leptokurtic. Also, since the probability value of obtaining the statistics is about 0.004771(0.4%) and it is less than the chosen level of significance (0.05 0r 5%), we reject the null hypothesis that the residuals are normally distributed.

To make the residual normal, we transformed the data by taking the difference of both variables and estimate the equation [d(Y) c d(X) ].  To check for normality, click from the result output VIEW>>RESIDUAL DIAGNOSTICS>>HISTOGRAM-NORMALITY TEST.



From the histogram graph, we see that the residuals are normally distributed. At the right hand corner we can see the J-B statistic, skewness, kurtosis and probability value. The J-B statistics is about 1.579047, the skewness of the residuals is closer to 0 (0.009905) and kurtosis is closer to 3 (2.744263). Also, since the probability value of obtaining the statistics is about 0.454061(45%) and it is greater than the chosen level of significance (0.05 0r 5%), we do reject the null hypothesis of normality in the residuals.


Dr. Jim Frost wrote ‘’ nonnormal residuals are not always a problem if you have a sample size of 100, you should be able to trust the F-test results". Read more on data transformation. Get more comment on why resduals need to be normal.

References


Dimitrios Asterious & Stephen G.Hal (2007). Applied econometrics: A modern approach. Palgrave Macmillan

Fisher, R. A. (1948). Conclusions fiduciaires. Ann. Inst. H. Poincaré 10. p. 191–213.

Jarque, C. M. & Bera, A. K. (1981). Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals: Monte Carlo Evidence. Economics Letters 7 (4).p

Gujarati D.N(2004). Basic Econometrics, 4th Edition. The McGraw−Hill Companies. p. 147-151

Lutkepohl H. & Kratzig M. (2004). Applied Time Series Econometrics. Cambridge University Press. p. 72-73.

Uğur ERGÜN & Ali Göksu (2013). Applied Econometrics With Eviews Applications. SarajevoI nternational Burch University. P 90-150.

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