Assignment #2: Hypothesis testing
Due February 14, 2014 in class
Obtain the annual data on CPI, nominal interest rates, nominal GDP and nominal private consumption in Canada from 1986 to current times following the steps in the data instructions file. First generate the real interest rate in Excel as long-term interest rate – inflation and call it realintrate.
Import the data in R. Draw a matrix of scatter plots in and report those: consumption against real interest rate, consumption against GDP. What do the scatter plots suggest about the relationship between consumption and two other variables?
As can be seen there is a positive almost perfect relationship between GDP and Consumption. The relationship between Consumption and Real interest rate does not seem linear – in earlier years positive and in later years negative.
For the model , where is real interest rate, is GDP, and is consumption, state your hypothesis regarding the signs and magnitudes (if applicable) of the coefficients based on what the economic theory suggests.
The Economic theory suggest that when the real interest rate increases, people save more and consume less. Therefore, we would expect a negative coefficient on the real interest rate. The theory does not suggest anything about the magnitude of the coefficient though. Therefore, for the real interest rate:
The coefficient on X2 is the marginal propensity to consume. The theory suggests that it must be positive and less than one. Therefore, and
Estimate the above equation by OLS (use R lm() command) and report your results. Interpret the meaning of each coefficient.
R-squared 0.9984 Adjusted R-squared 0.9983 F-test 11900 p-value 0 Dependent variable Coefficient Standard error t-statistics p-value
Constant -5830 3861 -1.51 0.139
GDP (beta 2) 0.554 0.0036 153.97 0.000
Real interest rate (beta 1) 1160 708 1.63 0.111 When GDP increases by $1mln. consumption increases by $554 thd. When the real interest rate increase by 1%, consumption increases by $1,160 mln.
How does the model perform? Explain what measures one can use to say something about a model’s fit and based on the regression output how well does your model fit the data?
The measure of the model’s fit is R-square. The closer the R-square to 1, the better the fit. You can see that R-square is 0.9984 – so it is almost a perfect fit.
Using the software regression output and the R linear.hypothesis() command to test the following hypotheses:
Report and explain your results (test statistics, decision rule).
The t-statistics for this test is 1.63 with the p-value of 0.111. Therefore, at 5% significance level we fail to reject the null that the coefficient is not significant.
This is just the t-test reported by R as part of the standard regression output. The t-stat for the test is 153.97 with the p-value of essentially 0. Therefore, at 5% significance level (and at 1% significance level as well), we reject the null hypothesis that the coefficient is insignificant (equals zero)
For this test, I have run the F test in R. The results are:
Linear hypothesis test
q1$GDP - q1$realintrate = 0
Model 1: restricted model
Model 2: q1$Consumption ~ q1$GDP + q1$realintrate Res.Df RSS Df Sum of Sq F Pr(>F)
1 39 4651182011
2 38 4347073461 1 304108550 2.6584 0.1113
So, the p-value of the test is 0.1113. Therefore, we fail to reject the null that the two coefficients are the same.
This hypothesis is the F-test that is reported as part of the standard output. The test statistics is F=11900 with the p-value of essentially zero. Therefore, at 5% significance level we reject the null that the model is not significant.
The test statistics for this hypothesis is t= -123.8925. This is a lower tail test with the p-value of 0.00. Therefore, at 5% significance level we reject the null in