Answer to Question #216410 in Microeconomics for israel Bereket

Question #216410

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


1
Expert's answer
2021-07-14T12:01:36-0400

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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