Answers: Problem Set 2
3.2 (a)
(b)
(c) See Figure 3.1.
(d) The quantities b1 and b2 are least squares estimates for the parameters b1 and b2 in the linear model where et is a random error term. The value b1 is an estimate of the intercept; it suggests that when xt = 0, yt = 1. The value b2 is an estimate of the slope; it suggests that when xt increases by 1 unit, the corresponding change in yt will be 1 unit.
(e) See Figure 3.1.
Y / X / Y_hat = 1 + 1*X5 / 3 / 4
2 / 2 / 3
3 / 1 / 2
2 / -1 / 0
-2 / 0 / 1
3.3 Show that the line passes through the point of means: the point of means is the ordered pair (X bar, Y bar) = (1,2). Show that the point (X=1,Y=2) is exactly on the line (has a residual of zero)
When X = 1, Y_hat is 2 and the actual value for Y is 2 à the residual is zero so the point lies on the line.
3.5 a) Use excel and follow the directions in Ch. 3 of the Excel supplement to get these answers:
(b) Strictly speaking, the intercept coefficient, b1 = 0.4656, gives the level of output when the feed input is zero. However, because a zero feed input is unrealistic, this interpretation must be treated with caution; it would be expected that, when the feed input is zero, output would be zero. The slope coefficient, b2 = 0.29246, indicates that, for a 1 unit change in the feed input, there will be a corresponding change of 0.29 in the poultry food output.
c) the estimated production function is the straight line in the graph below.
d) skip
3.9 (a) The model is a simple regression model because it can be written as where y = rj - rf , x = rm - rf , b1 = aj and b2 = bj.
(b) The estimate of Mobil Oil's beta is b2 = 0.7147. This value suggests Mobil Oil's stock is defensive.
(c) b1 = = 0.00424. This small value seems consistent with finance theory.
3.11 The observations on y and x and the least-squares line estimated in part (b) are graphed in Figure 3.8. The line drawn for part (a) will depend on each student’s subjective choice about the position of the line. For this reason, it has been omitted.
(b) Preliminary calculations yield:
The least squares estimates are
Figure 3.8 Observations and Fitted Line for Exercise 3.11
Figure 3.9 Plot of residuals against x for Exercise 3.11
(c)
The predicted value for y at is
We observe that That is, the predicted value at the sample mean is the sample mean of the dependent variable . The least-squares estimated line passes through the point .
(d) To get the least squares residuals, we must first computer the predicted values for y:
when x = 1,
when x = 2,
when x = 3,
when x = 4,
when x = 5,
when x = 6,
The residuals are :
y / y_hat / e_hat4 / 4.1904 / -0.1904
6 / 5.4475 / 0.5525
7 / 6.7046 / 0.2954
7 / 7.9617 / -0.9617
9 / 9.2188 / -0.2188
11 / 10.4759 / 0.5241
Their sum is
(d) The least-squares residuals are plotted against x in Figure 3.9. They seem to exhibit a cyclical pattern. However, with only six points it is difficult to draw any general conclusions.
3.14 We are given the values for Xbar and Ybar which we know will lie on the estimated line. Let this point of means be the point Xo, Yhato on the line.
We are also given one additional point on the estimated line: and . Recall from algebra class that, given two points that lie on the same line, we can back out the slope and intercept and thus express the line as an equation. The slope
The intercept: à the estimate line is
3.15 (a) The intercept estimate is an estimate of the number of sodas sold when the temperature is 0 degrees Fahrenheit. A common problem when interpreting the estimated intercept is that we often do not have any data points near If we have no observations in the region where temperature is 0, then the estimated relationship may not be a good approximation to reality in that region. Clearly, it is impossible to sell -240 sodas and so this estimate should not be accepted as a sensible one.
The slope estimate is an estimate of the increase in sodas sold when temperature increases by 1 Fahrenheit degree. This estimate does make sense. One would expect the number of sodas sold to increase as temperature increases.
(b) If temperature is 80 degrees, the predicted number of sodas sold is
(c) If no sodas are sold, and
or .
Thus, she predicts no sodas will be sold below 40°F.
(b) A graph of the estimated regression line appears below
Graph of regression line for soda sales and temperature for Exercise 3.15
3.16 Use the data in the file br-1.dat to estimate the model in Excel following the instructions from Chapter 3 of the Excel supplement. The results should be:
(a)
Variable / Mean / St. Devn. / Minimum / Maximumprice / 82133 / 12288 / 52000 / 111002
sqft / 1794.6 / 200.02 / 1402.0 / 2100.0
(b) The plot in Figure 3.15 suggests a positive relationship between house price and size.
(c) The estimated equation is
Price_hat = -426.71 + 46.005sqft
The coefficient 46.005 suggests house price increases by approximately $46 for each additional square foot of house size. The intercept, if taken literally, suggests a house with zero square feet would cost $-426. The model should not be accepted as a serious one in the region of zero square feet, a meaningless value.
The excel output is:
(d) A plot of the residuals against square feet appears below. There appears to be a slight tendency for the magnitude of the residuals to increase for larger-sized houses.
(e) The predicted price of a house with 2000 square feet of living space is
price_hat = -426.71 + 46.005(2000) = 91,583
(f) This model predicts that an increase in sqft of 1 à price increases by $46, so an increase of 1000 square feet is predicted in increase price by 46.005(1000) = $46,005