# Interpreting Panel Data Regression Models: An Annotated Stata Output

Interpreting Panel Data Regression Models is commonly requested by many students on social media groups or through private messages on Facebook and LinkedIn. In this tutorial, I will elaborate how to interpret different results from Panel Data Regression Models like Fixed Effects and Random Effects models. You can learn Panel Data Analysis in private course with us.

Interpreting Panel Data Regression Models

In this section, we are Interpreting Panel Data Regression Models with Fixed Effects.

We can see the left top corner of the first set of regression results that Number of obs = 3,792 which is actually time period (t) multiplied by entities (i). The number of groups (i) in this data is 130. So we can find average number of time period for which these (i) has data is 3,792/130 = 30 (29.XX). This does not mean the maximum number of time periods for which data is available for a given variable, say debt in our example. There can be some variables for some countries where data can be available for less than 30 years and for some countries the same variable will have more than 30 years of data. So the value of 30 means on average 30 years of data is available for countries. This can be seen from the the same part of panel data fixed effects regression given in the left top corner of the output in Stata. We can see minimum observation per (i) is 2, maximum is 36 and on average it is 36.

The lower part of this section of the output shows the F test which is loosely considers goodness of fit statistic (in addition to R Square statistics or explanatory power of the regression model). So if the F statistic here is greater than the tabulated value of F distribution for given degrees of freedom, we will reject the null hypothesis that all the included independent variables (Xit = gdp gfc indva trade) does not affect/determine the dependent variable (y = debt). We can see the the degrees of freedom include a value of 4 which is equal to the Xit so we can guess from this it the k parameters of regression. The second value is (N-i-k = 3,792-130-4). If the p-value corresponding to this test is less than 0.05, we reject the null hypothesis that all the Xit variables do not affect Yit and accept the alternative that all Xit affect Yit (or at least some of them do).

The second half of the top results include R square. As we could see from the panel structure, the variations in Yit can be both at time based on entity based. So we can split the total R squared into two types. Within R square and Between R Square. It should be clear that R square in Panel Data models is not simple R squares that we obtain from OLS estimators. These R square values are based on correlations actually between the actual Yit and its Predicted values from the regression equation.

Let us assume, when we estimate a panel data model and its prediction system becomes like this:

The predictions from equation 1 and its correlation with actual Y gives overall R square. The prediction from equation 2 and the actual Y gives Between R square while the predictions from equation 3 and actual Y gives Within R squares. We can see that within R square is = 0.9710, between R square is = 0.9940 and overall R square is = 0.9764 and these are relatively higher indicators of goodness of fit statistic

The last section of the top results is an indication of the correlation between entity specific error term as Ui and Xb which is model explanatory variables. We can from the above results output that there is a high correlation between Ui and Xb which is equal to 0.6631. We can see that a value close to 0 is a weaker indicator of no fixed effects in the model and hence we might need an alternative specification for panel data. This is not a formal test but provides a basis for further formal tests to compare Fixed Effects models with Random Effects Models like we do in Hausman test that we will explain in a separate post.

Before reading the next paragraph, please note that main table of coefficients and other statistics is given after the above statistics at top. The first column in this table is usually the Yit and the other variables are all Xit in the model in rows. The Intercept of the regression model in Stata is denoted by _cons.

The key issue in Interpreting Panel Data Regression Models is related to the above few bits of statistics that commonly we miss to understand. The key results all by the way, the table with coefficients (in Stata results it is denoted as Coef.) , standard error (in Stata results it is denoted as Std. Err.), t or z statistics (in Stata results it is denoted as t or Z), p-values(in Stata results it is denoted as P>|t|) and confidence intervals (in Stata results it is denoted as [95% Conf. Interval]). I would explain these in turn in a single line each. Coefficient is the effect of Xit on Yit and we have to note its sign and size both. The objective of regression model determines if we need sign or size or both. For inferences, we usually need the sign only, for predictive analysis, we do need sign and for both we have to use sign and size of the coefficient. Also, the variation in Yit is both across time and i so we have to denote the effect of Xit on Yit in that context as well. So the interpretation of Coef. is in terms of time and i both.

The second column is Standard Errors of the coefficients. These are the standard deviations from many coefficients of the same variable in a similar regression model estimated from various samples taken from the same data. We call this process as sampling distribution. Thus, the standard errors shows us the margin of error within which the coefficients can vary across different cases. The standard errors are required to compute the t statistics or Z statistics we will use for hypothesis testing on coefficients to determine the significance of effects.

The next column shows the values of t statistics for all coefficients. These are interpreted simply to showcase if the given coefficient is statistically significant or not. The null hypothesis on coefficients are stated as: Ho: b(X)=0 against the alternative that H1: b(0) is not equal to 0. If the value of t statistic is greater than 1.96, we reject the null hypothesis and accepts the alternative hypothesis. Thus, rejection of the null hypothesis based on the observation that value of t statistic is greater than 1.96 means the effect of X is significantly different than 0 or simply putting, we can say X affects Y significantly. This should be stated for each independent variable in the model. The null hypothesis on intercept can be interpreted as the values of Y is different than 0 when all the X variables are joinly 0. This does not test any effect by the way.

The most important results in Interpreting Panel Data Regression Models is p-value. The next column in Fixed Regression results in the given Stata output shows the p-values. If the p-value is less than 0.05, we reject the null hypothesis that Ho: b(X)=0 against the alternative that H1: b(X) is not equal to 0.

It is the average….

In this section, we are Interpreting Panel Data Regression Models with Random Effects.

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