Quiz 1
Consider the following regression results
LnY= -3.32
+1.5LnX2
+0.5LnX3
SE= (2.4559)
(0.5415)
(0.1041)
R2=0.88, Adj R2=0.87, DF=12, T= (/2,12) = 2.179
Y= real output
X2= labor input
X3= capital input
1. Interpret this regression model.
2. Test the hypothesis that there is no relationship between real output and labor input.
3. Test the significance of capital on real output.
4. Build a 95% confidence interval for labor input (b2) and capital input(b3).
Solution
1.
We have fitted a linear regression model to the Natural logarithm of the value of real output on the
values of labor input and capital input.
It is very important to keep in mind that the linear relationship has been defined using logarithm
and not normal values.
The intercept of the model takes the value -3.32. This means that when the logarithm of labor
input and capital input is 0, the logarithm of real output is -3.32.
The coefficient of logarithm of labor input is 1.5, This means that as the logarithm of labor
inputs changes by 1 unit, the logarithm of real output changes by 1.5 units in the same direction
(since the value of coefficient is positive)
The coefficient of logarithm of capital input is 0.5, This means that as the logarithm of capital
inputs changes by 1 unit, the logarithm of real output changes by 0.5 units in the same direction
(since the value of coefficient is positive)
The R^2 and Adj R^2 of the model is high, ie,0.88 and 0.87 respectively. This means that the
model is a pretty good fit, as it explains a very high proportion of variation in the data. The
degree of freedom is 12 (which is n-2 where n is the sample size), meaning that the size of the
sample using which the multiple regression model has been estimated is 12+2=14, i.e., n=14.
2.
Let b2 denote the coefficient for the labor input.
If there is a relation between labor input and real output, we would expect that the true value of
coefficient b2 is not equal to 0, hence meaning that labor input has an impact on the value of the real
output.
Let us check whether b2 is statistically significant or not. The hypothesis for the test is given as
H0: b2=0, meaning there is no relationship between labor input and real output.
H1: b20, meaning there is a relationship between labor input and real output.
The test statistics is given by:
Tc =
To calculate the test statistic, we assume that the null hypothesis is true. Therefore, b2=0 Hence the test
statistic is given as
1.5-0/0.5415=2.770083
The critical value has been given in the question in the form of t(/2, 12)=2.179 Since the test statistic is
greater than the critical value, we have sufficient evidence to reject the null hypothesis. Therefore, we
reject the null hypothesis and conclude that there is a relationship between the labor input and real
output.
3.
On similar lines we carry out the test to check whether there is any significant relationship between
capital input and real output.
Let us check whether b3 is statistically significant or not. The hypothesis for the test is given as
H0: b3=0, meaning there is no relationship between labor input and real output.
H1: b30, meaning there is a relationship between labor input and real output.
The test statistics is given by:
Tc =
To calculate the test statistic, we assume that the null hypothesis is true. Therefore b3=0 Hence the test
statistic is given as
0.5-0/0.1041=4.803074
The critical value has been given in the question in the form of t(/2, 12)=2.179 Since the test statistic is
greater than the critical value, we have sufficient evidence to reject the null hypothesis. Therefore, we
reject the null hypothesis and conclude that there is a relationship between the capital input and real
output.
4.
The 95% confidence interval for a regression variable (b) is given as
𝑏 ± 𝑡(
,
)
× 𝑠𝑒(
)
Confidence interval (LI) =1.5 + 2.179*0.5415
Ci= 0.3200715<b2<2.6799285
Confidence interval (ci) =0.5 + 2.179*0.1041
Ci= 0.2731661<b2<0.7268339
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