Download Interpreting the Coefficients of a Regression Analysis: A Trade Balance Example and more Summaries Economics in PDF only on Docsity!
Economics 310 Menzie D. Chinn
Fall 2004 Social Sciences 7418
University of Wisconsin-Madison
Problem Set 5 Answers
This problem set is due in lecture on Wednesday, December 15th. No late problem sets will be
accepted. Be sure to show your work (that is, do not use a spreadsheet or statistical program to
generate your answers), and to write your name, ID number, as well as the name of your
Teaching Assistant, on your problem set.
Answer all these problems. They are from the textbook, with the exception of Problem W which
is written out.
Problem W. Below are plotted data for the trade balance as a share of GDP for the US and US GDP.
The idea is that when the US economy booms, imports rise, and the trade balance deteriorates.
-.
-.
-.
-.
-.
-.
.
.
.
8.2 8.4 8.6 8.8 9.0 9.2 9. LOGGDP00US
TB_RATIO
TB_RATIO vs. LOGGDP00US
Figure 1: Time series for Trade Balance to GDP Figure 2: Scatter plot of Trade Balance
Ratio and US Real GDP. Sources: BEA. Ratio and US Real GDP.
Dependent Variable: TB_RATIO Method: Least Squares Date: 12/07/04 Time: 20: Sample: 1973:1 2004: Included observations: 126
Variable Coefficient Std. Error t-Statistic Prob.
C 0.302216 0.027757 10.88792 0. LOGGDP00US -0.036190 0.003152 -11.48326 0.
R-squared 0.515370 Mean dependent var -0. Adjusted R-squared 0.511462 S.D. dependent var 0. S.E. of regression 0.009867 Akaike info criterion -6. Sum squared resid 0.012071 Schwarz criterion -6. Log likelihood 404.1663 F-statistic 131. Durbin-Watson stat 0.099030 Prob(F-statistic) 0.
a. In words, interpret the coefficient on LOGGDP00US (where this is the log of GDP00US ).
Each one percent change in real US GDP causes a 0.036 percentage point decrease in the trade balance to GDP ratio. Technically, it is ∆ tb / ∆ y where tb is the trade balance to GDP ratio and y is log real GDP.
b. Conduct a one-sided t-test using a 1% significance level, for the following hypothesis test:
H 0 : β 1 = 0
HA: β 1 < 0
-.
-.
-.
-.
-.
-.
.
.
.
1975 1980 1985 1990 1995 2000 TB_RATIO LOG(GDP00US)
US Trade Balance to GDP ratio US Real GDP (2000C$)
t =
The critical value at 1% level for a one tailed test, using 120 degrees of freedom (actually there are 124 d.f.), is -2.358. Since -11.31 < -2.358, reject the null hypothesis.
c. Calculate the standard error of the regression, using the statistics reported in the output (show your
work!).
s
SSR
n
. which matches the entry under “S.E. of regression”.
d. Calculate the value of the “S.E. of regression” using the “Sum of squared resid” (also termed the
Sum of Squared Errors in the textbook).
s
SSR
n
. which matches the entry under “S.E. of regression”.
e. Calculate the R-squared using the values for SSE and the “S.D. dependent var” (which is the
standard deviation of the dependent variable).
R 2 = 1 – (SSE/SS (^) yy). While SS (^) yy is not reported in the regression output, notice that the standard deviation of the dependent variable y is reported. In fact SS (^) yy = (SD 2 )x(n-1) = (0.014116)^2 x(125) = 0.
Hence, R 2 = 1 – (SSE/SS (^) yy) = 1 – (0.012071/0.024907682) = 0.515370 (which matches the regression output entry for this measure).
The above story omits some other relevant factors. When the rest-of-the-world booms, then their
imports (our exports) rise, so the rest-of-world GDP ( GDP96_ROW ) should enter. But what should
also matter is the real value of the US dollar ( DOLLAR_FEDBROAD ), which determines how
expensive American goods are relative to foreign. Figures 3 and 4 depict the trade balance and the
dollar value (in logs).
-.
-.
-.
-.
-.
-.
.
.
.
4.4 4.5 4.6 4.7 4.8 4. LOGDOLLAR_FEDBROAD
TB_RATIO
TB_RATIO vs. LOGDOLLAR_FEDBROAD
-.
-.
-.
-.
-.
-.
.
.
.
1970 1975 1980 1985 1990 1995 2000 TB_RATIO LOG(DOLLAR_FEDBROAD)
Trade Balance to GDP ratio
Log Real Value of US$
The confidence interval is given by β$^3 ± t α / 2 × s β 3 = -0.064503 ± 1.98 x 0.
= (-0.077521, -0.051512`)
j. If the dollar were to depreciate by 20%, what would happen to the trade balance, and by how much?
∆ tb ∆ tb r = ⎛⎝⎜ ⎞⎠⎟ × r
where r is the log real value of the dollar.
∆ tb = - 0.064503 × − 0 20. =0 0129. or in other words, the trade balance to GDP ratio improves by 1.3 percentage points.
9.12.2004 e310ps5a_f
