The preceding articles showed how to conduct time series analysis in STATA on a range of univariate and multivariate models including ARIMA, VAR (Lag selection, and stationarity in VAR with three variables in STATA) and VECM (VECM in STATA for two cointegrating equations). Time series data requires some diagnostic tests in order to check the properties of the independent variables. This is called ‘normality’. This article explains how to perform a normality test in STATA.
The normality test helps to determine how likely it is for a random variable underlying the data set to be normally distributed. There are several normality tests such as the Skewness Kurtosis test, the Jarque Bera test, the Shapiro Wilk test, the Kolmogorov-Smirnov test, and the Chen-Shapiro test. This article shows two tests; Skewness Kurtosis and Jarque Bera tests because they are simple and popular.
This article uses quarterly data of the following variables of the Indian economy for the time period 1997- 2018:
- Gross Domestic Product (GDP) (dependent variable).
- Gross Fixed Capital Formation (GFC) (independent variable).
- Private Final Consumption Expenditure (PFC) (independent variable).
The first step is to set the time series data. Thereafter proceed to the regression analysis.
The time series result will identify the residuals from the regression analysis.
Next, use the below command in order to generate the residuals in the data set.
regress gdp gfcf pcfe
In order to predict the residuals from the regression model, use the below command.
predict resid, residuals
The command ‘predict’ helps in generating new variables (here it is resid or the error term in the model). This will create a new variable ‘resid’ in the data editor (figure below).
Conducting a normality test in STATA
In order to generate the distribution plots of the residuals, follow these steps (figure below):
- Go to the ‘Statistics’ on the main window
- Choose ‘Distributional plots and tests’
- Select ‘Skewness and kurtosis normality tests’.
After performing the above procedure, ‘sktest – Skewness and kurtosis test for normality’ box will appear (figure below). Select the main variable to test for normality (here it is ‘resid’).
This will give the following results (figure below). The null and alternative hypotheses for the normality test are:
- Null hypothesis: The data follows a normal distribution.
- Alternative hypothesis: The data does not follow a normal distribution.
Skewness Kurtosis test for normality
Skewness is a measure of the asymmetry of the probability distribution of a random variable about its mean. It represents the amount and direction of skew. On the other hand, Kurtosis represents the height and sharpness of the central peak relative to that of a standard bell curve. The figure below shows the results obtained after performing the Skewness and Kurtosis test for normality in STATA.
‘sktest’ shows the number of observations (which is 84 here) and the probability of skewness which is 0.8035 implying that skewness is asymptotically normally distributed (p-value of skewness > 0.05). Similarly, Pr(Kurtosis) indicates that kurtosis is also asymptotically distributed (p-value of kurtosis > 0.05). Finally, chi(2) is 0.1426 which is greater than 0.05 implying its significance at a 5% level. Consequently, the null hypothesis cannot be rejected. Therefore, according to the Skewness test for normality, residuals show normal distribution.
Jarque Bera test for normality
The other test of normality is the Jarque Bera test. In order to perform this test, use the command ‘jb resid’ in the command prompt.
The results will appear (figure below).
If the p-value is lower than the Chi(2) value then the null hypothesis cannot be rejected. Therefore residuals are normality distributed.
As per the above figure, chi(2) is 0.1211 which is greater than 0.05. Therefore, the null hypothesis cannot be rejected. Moreover, there is no violation of the normal distribution assumption of error terms as the residuals are coming out to be normal.
Normality through histogram
A histogram plot also indicates the normality of residuals. A bell-shaped curve shows the normal distribution of the series. In order to generate the histogram plot, follow the below procedure.
- Go to ‘Graphics’ in the main bar.
- Select ‘histogram’.
The below figure will appear.
Then choose the main variable and choose ‘Density’ under the Y-axis section. Click on ‘OK’ (figure below).
The below window will appear. Click on ‘Add normal density plot’.
Finally, click on ‘OK’ to generate the histogram plot showing the normality distribution of the residuals (figure below).
The figure above shows a bell-shaped distribution of the residuals. The X-axis shows the residuals, whereas Y-axis represents the density of the data set. Thus this histogram plot confirms the normality test results from the two tests in this article.
The next article discusses the tests for heteroscedasticity. Heteroscedasticity is a violation of an important ordinary least squares (OLS) assumption that all residuals belong to a population that has a constant variance (homoscedasticity).