POIR 611
Graham
Homework #3
Part 1: Gujarati
Chapter 2: Problems 2.3, 2.7, 2.10
Chapter 3: Problems 3.2, 3.14, 3.19
Chapter 2
2.3. What is the role of the stochastic error term ui in regression analysis? What is the difference
between the stochastic error term and the residual, ui hat?
- Stochastic error term: random, nonsystematic term, a random “disturbance,” the effect of
the variables that were omitted from the equation, assumed to have a mean value of zero,
and to be uncorrelated with the independent variable, x, assumed to have a constant
variance, and to be uncorrelated with its own past values (i.e., it is “noise”)
- Residual: estimated error, the vertical distance between the actual value of the dependent
variable, yi, and the fitted value, yˆi = βˆ0 − βˆ1xi → uˆi = yi − yˆi = yi − βˆ0 − βˆ1xi
2.7. Are the following models linear regression models? Why or why not?
- Exponential functions (y =ex), logarithmic functions (x = by, y = logb(x) ), power functions
(f(x) = cxa) are nonlinear
- (a) If we take the natural log, Yi = b 1 + b 2 Xi + ui, becomes a linear regression model.
- (b) If we transform the equation (using the logit transformation), ln [(1- Yi)/Yi] = b 1 + b 2
Xi + ui becomes a linear regression
- (c) linear
- (d) nonlinear
- (e) nonlinear
2.10. You are given the scattergram along with the regression line. What general conclusion do
you draw from this diagram? Is the regression line sketched in the diagram a population
regression line or the sample regression line?
- There is a weak, positive, linear relationship between the average annual change in
export-GNP ratio and the average growth in real manufacturing wages in % per year
- The regression line sketched is a sample regression line based on 50 developing countries
Chapter 3
3.2. Show that the estimates are in fact the OLS estimators
- OLS estimates are obtained from the sample data, and the regression line passes through
̅)
the sample means of Y and X. So this means that ̅
Y = β̂1 + β̂2 (X
̅) → (28/4) = (1.572) + (1.357) (16/4) → 7 = 7
- From the table, ̅
Y = β̂1 + β̂2 (X
3.14. In the regression suppose we multiply each X value by a constant. Will it change the
residuals and fitted values of Y? What if we add a constant value to each X value?
- The residuals and fitted values of Y will not change.
- Let Yi = b 1+ b 2Xi + ui and Yi = a1+a 2Zi + ui , where Z = 2X. We know that
-
-
and
Which means that
That is the intercept term remains unaffected. As a result, the fitted Y values and the
residuals remain the same even if Xi is multiplied by 2. The analysis is analogous if a
constant is added to Xi.
3.19. Y = exchange rate (CD/$), X = ratio of US CPI / Canadian CPI = relative prices
Yhat = -0.912 + 2.250 X, r2 = 0.440
a. One unit of increase in the ratio of US CPI / Canadian CPI leads to 2.250 increase in the
exchange rate (CD/$). The squared coefficient of correlation shows that the regression
line is somewhat of a good fit of the data points.
b. Larger the CPI, higher the inflation. X value larger than 1 means that inflation in the US
is greater than inflation in Canada. According to the regression line, the higher inflation
in the US relative to Canada, higher the exchange rate of CD to $. Higher the inflation,
weaker the currency. Lower the inflation, stronger the currency. So it follows that higher
inflation in the US would be related to stronger the Canadian dollar.
c. If we redefine X as Canadian CPI / US CPI, then the sign would be changed to negative.
This is because lower inflation in Canada will lead to weaker US dollar.