Suppose that a researcher estimates a consumption function and obtains the
following results:
C=15+0.81Yd n=19 R2=0.99
(3.1) (18.7)
where C=Consumption, Yd=disposable income, and numbers in the parenthesis are the
‘t-ratios’
a. Test the significant of Yd statistically using t-ratios
b. Determine the estimated standard deviations of the parameter estimates
a
For a fitted regression model "Y=\\hat{\\beta_1}+\\hat{\\beta_2}x" with Y as response and X as prediction variable, the test statistic for testing the significance of X is given by,
"T=\\frac{\\hat{\\beta_2}-{\\beta^0_2}}{\\sqrt{\\frac{\\hat{\\sigma^2}}{s_{xx}}}}" "\\utilde{H_0}\\space t_n-2"
where, "\\hat{\\sigma^2}=\\frac{Rss}{n-2}, s_{xx}=\\displaystyle\\sum_{i=1}^n(x_i-\\bar{x})^2, \\bar{x}=\\frac{1}{n}\\displaystyle\\sum_{i=1}^nxi,"
"\\beta^0_2" is the hypothesized value of "\\beta_2," here it is 0.
a.
In this case, "Y=C,x=Y_d, n=19, R^2=0.99"
T-ratio for "\\beta_1=3.1," T-ratio for "\\beta_2=18.7"
Therefore we are to test the null hypothesis, "H_1:\\beta_2=0."
against the alternative hypothesis, "H_1:\\beta_2\\not=0."
The test statistic for this test is given by, "t_{\\beta_2}=" T-ratio for "\\beta_2=18.7" [ ∵ under "H_0, \\beta^0_2=0]"
The p-value for this test can be computed for t-distribution with "f=n-2=19-2=17" using R code:
"2^Hpt(18.7,17,lower.tail=F)"
which gives p-value as 0 for which we reject the null hypothesis.
∵ we can conclude that "Y_d" is statistically significant.
b.
The t-ratio is basically the estimate divided by the standard error. Again the standard error is the standard deviation of the estimates.
∵ "t-ratio(\\beta_1)=3.1=\\frac{\\hat{\\beta^1}}{se\\hat({\\beta_1})}=\\frac{15}{se\\hat({\\beta_1})}"
"\\implies se({\\beta_1})=\\frac{15}{3.1}=4.839"
"t-ratio(\\beta_2)=18.7=\\frac{\\hat{\\beta^2}}{se\\hat({\\beta_2})}=\\frac{0.81}{se\\hat({\\beta_2})}"
"\\implies se({\\beta_2})=\\frac{0.81}{18.7}=0.043"
Therefore the estimated standard deviations of the parameter estimates are 4.389 and 0.043 respectively.
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