Question

Please explain the Significance F and the p-value of this multiple regression.

Variable 1: Quantity of new domestic car sales

Variable 2: Federal funds rate

Outcome: Demand for money

Regression Statistics
Multiple R	0.978788301
R Square	0.958026537
Adjusted R Square	0.957895165
Standard Error	107.4145035
Observations	642
ANOVA
	df	SS	MS	F	Significance F
Regression	2	168278816.1	84139408.05	7292.452378	0
Residual	639	7372702.481	11537.87556
Total	641	175651518.6
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	59.28318528	12.38080726	4.788313397	2.09203E-06	34.97119982	83.59517075	34.97119982	83.59517075
X Variable 1	-28.45515209	1.258525275	-22.60991706	1.29925E-83	-30.92649726	-25.98380692	-30.92649726	-25.98380692
X Variable 2	2.571422557	0.023723808	108.3899603	0	2.52483651	2.618008604	2.52483651	2.618008604

Answer 1

The test statistic (t stat, F) and p-value/ Significance F are always tied to the hypothesis we are testing. They do not have meaning outside this. Hence we will look at the specific hypotheses and when to use these statistic.

Using the above ANOVA table

The test statistic is

F=7292.452

note: To calculate this we need to know mean square regression MSR=84139408.05 and means square error/residual, MSE=11537.87. We also know that the ratio MSR/MSE has F distribution.

Then the F statistic is

The p-value is calculated as  

The numerator degrees of freedom df=2 and denominator df=639 (all from the above table)

We can use calculator or Excel function =F.DIST.RT(7292.452,2,639) to get the above probability.

This is indicated as "Significance F" in the ANOVA table above.

the p-value=0.0000

The p-value/significance F indicates the probability of observing this F statistic (and hence this regression model output) if the null hypothesis H₀ given above is true (which is the slopes are both =0). A low p-value indicates that the null hypothesis may not be true.

We will reject the null hypothesis, if the p-value is less than the significance level alpha=0.05. That means o

Here, the p-value is 0.0000 and it is less than 0.05. Hence we reject the null hypothesis.

ans: We reject the null hypothesis. The regression model is significant in explaining Demand for money.

We know that the sampling distribution of estimated slope/coefficient has t distribution. Hence we calculate a t statistic.

To calculate the test statistic to test the hypotheses, we need the estimated value of the slope

and the standard error of slope

Next the hypothesized value of slope from the null hypothesis is . That is we assume that the true value slope is 0 and hence the mean of the distribution of slope is 0.

The test statistic is

This is the value of t stat given in the output above. Note that the t stat given in the output is valid only to test the hypothesis that slope of X1 is 0. However, let us say you want to test that the slope of X1 is equal to say -25, then you know that the above t stat cannot be used.

Hence t stat is just the standardized value of estimated value of slope (which is -28.4552 in this case) when the null hypothesis is true (which is the slope is 0 in this case)

Next we need the p-value. Note that this is a 2 tailed test as the alternative  hypothesis has "not equal to".

The p-value is

Number of observations used is n=642 and there are k=2 independent variables. The degrees of freedom=n-k-1=642-2-1=639.

We can use calculator or Excel function =T.DIST.2T(22.6099,639) and get 1.29925E-83

From the output we get the p-value =0.0000

The p-value indicates the probability of observing this slope estimate of -28.4552 if the null hypothesis H₀ given above is true (which is the slope of X1 is 0). A low p-value indicates that the null hypothesis may not be true

We will reject the null hypothesis, if the p-value is less than the significance level alpha=0.05.

Here, the p-value is 0.0000 and it is less than 0.05. Hence we reject the null hypothesis.

ans: We reject the null hypothesis. The slope coefficient of X1 is significant. The variable Quantity of new domestic car sales is able to significantly explain the demand for money

We can do the same thing for the next coefficient.

To calculate the test statistic to test the hypotheses, we need the estimated value of the slope

and the standard error of slope

Next the hypothesized value of slope from the null hypothesis is

The test statistic is

the test statistic is t=108.39. We confirm this from the output given above.

Hence t stat is just the standardized value of estimated value of slope, 2.5714 assuming that the mean of the slope estimate is 0, which is when the null hypothesis is true (which is the slope is 0 in this case)

Next we need the p-value. Note that this is a 2 tailed test as the alternative  hypothesis has "not equal to".

The p-value is

Number of observations used is n=642 and there are k=2 independent variables. The degrees of freedom=n-k-1=642-2-1=639.

We can use calculator or Excel function =T.DIST.2T(108.39,639) and get 0

From the output we get the p-value =0.0000

The p-value=0.0000

We will reject the null hypothesis, if the p-value is less than the significance level alpha=0.05.

Here, the p-value is 0.0000 and it is less than 0.05. Hence we reject the null hypothesis.

ans: We reject the null hypothesis. The slope coefficient of X2 is significant. The variable Federal funds rate is able to significantly explain the demand for money