EC260 Lecture Notes - Lecture 3: Ordinary Least Squares, Null Hypothesis, F Communications

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24 Aug 2016

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Y = b0 + b1x1 + b2x2 + . bkxk + u. Intuitively, ols is fitting a line through the sample points such that the sum of squared residuals is as small as possible (hence least squares) Residual, , is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and sample point. Sample regression line, sample data points, and associated estimated error terms: Sample regression line can be written as: y i. 1 x i estimator of the population mean e(yi) Estimator of the intercept of the population regression ( Estimator of the slope of the population regression line ( 2 e i y i y i estimated error/residual. Least squares estimates vary from sample to sample b/c you obtain data from different places in each sample. The probability distribution of the least squares estimators. Regarded as random variables, the least squares estimators distributions.

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TABLE 66 Stock Prices and Consumer Prices

CITY
Y = Rate of Change, Stock Prices, Percent Per Year
X = Rate of Change, Consumer Prices, Percent Per Year

CITY	Y	X
A	5	4.3
B	11.1	4.6
C	3.2	2.4
D	7.9	2.4
E	25.5	26.4
F	3.8	4.2
G	11.1	5.5
H	9.9	4.7
I	3.3	2.2
J	1.5	4
K	6.4	4
L	8.9	8.4
M	8.1	3.3
N	13.5	4.7
O	4.7	5.2
P	7.5	3.6
Q	4.73.	6
R	8	4
S	7.5	3.9
T	9	2.1

Table 66 gives data on percent change per year stock prices (Y) and consumer prices (X) for a cross section of 20 cities.

******************* answer in "SAS format" please********************* (if possible)

1) Plot the data in scattergram

2) Regress Y on X and examine the residuals from this regression. What do you observe?

3) Since the data for city(E) is unusual, repeat the regression in (2) dropping the data on city(E). Now examine the residuals from this regression. What do you observe?

4) If on the basis of the results in (2) you conclude that there was heteroscedasticity in the error variance but on the basis of the results in (3) you reverse your conclusion, what general conclusions do you draw?

State whether the following statements are true or false. Breifly justify your answer:

5) When autocorrelation is present, OLS estimators are biased as well as inefficient;

6) The R squared values of two models, one involving regression in the first-difference form and another in the level form, are not directly comparable.

7) In the presence of heterscedasticity the usual OLS method always overestimates the standard errors of estimators.

8) If a regression model is mis-specified (e.g., an important variable is ommitted), the OLS residuals will show a distinct pattern.

redcamel374

1. You are given only three quarterly seasonal indices and quarterly seasonally adjusted data for the entire year. What is the raw data value for Q4? Raw data is not adjusted for seasonality.

Quarter Seasonal Index Seasonally Adjusted Data

Q1 .80 295

Q2 .85 299

Q3 1.15 270

Q4 --- 271

2. One model of exponential smoothing will provide almost the same forecast as a liner trend method. What are linear trend intercept and slope counterparts for exponential smoothing?

A. Alpha and Delta

B. Delta and Gamma

C. Alpha and Gamma

D. Standard Deviation and Mean

3. When performing correlation analysis what is the null hypothesis? What measure in Minitab is used to test it and to be 95% confident in the significance of correlation coefficient.

A. Ho: r = .05 p < .5

B. Ho: r = 0 p >.05

C. Ho: r ? 0 p?.05

D. Ho: r = 0 p?.05

In decomposition what does the cycle factor (CF) of .80 represent for a monthly forecast estimate of a Y variable?

A. The estimated value is 80% of the average monthly seasonal estimate.
B. The estimate is .80 of the forecasted Y trend value.
C. The estimated value is .80 of the historical average CMA values.
D. The estimated value has 20% more variation than the average historical Y data values.

5. A Wendy's franchise owner notes that the sales per store has fallen below the stated national Wendy's outlet average of $1,368,000. He asserts a change has occurred that reduced the fast food eating habits of Americans. What is his hypothesis (H1) and what type of test for significance must be applied?

A. H1: u ? $1,368,000 A one-tailed t-test to the left.
B. H1: u = $1,368,000 A two-tailed t-test.
C. H1: u < $1,368,000 A one-tailed t-test to the left.
D. H1: p < $1,368,000 A one-tailed test to the right

6. As the sample size from a population increases, for a given level of significance what happens to the null hypothesis rejection region and size of the t-table value?

A. The rejection region and the t-table value generally gets smaller for sample size below 31.
B. The rejection region gets larger and the t-table value generally gets smaller for sample sizes below 31.
C. The rejection region remains unchanged while the t-table value gets smaller for all sample sizes.
D. The zero mean hypothesis region gets larger and the t-table value gets larger as well for sample sizes below 31.

7. You obtained autocorrelation LBQ value of 18.58 for the 12^th lag from a data sample. Are the data significantly correlated at the lag examined or not? You want to be 95% confident in your answer.

A. Yes. The data are significantly correlated through the 12th lag.
B. No. The data are not significantly correlated through the 12^th lag.

C. No. Only the 12 lag period is not correlated.

D. You cannot tell since the number of sample observations is not provided.

E. The p-value is above .05 so the data is correlated.

8. Sometimes forecasters get lazy or forgetful and do not check the significance of XY data correlations and use the X variable to forecast Y. What is the result of this?

A. Type 2 error
B. Autocorrelation error
C. Type 3 error
D. Type 1 error

9. Do error measures indicate the statistical significance of forecast model variables?

A. Yes. They move in the same direction as statistical significance.
B. Yes. As error measures decrease the variable significance increases.
C. No They indicate only the magnitude of estimate error.
D. No. They indicate only the statistical significance of a forecast or fitted values.

10. In exponential smoothing what is the weight of the alpha coefficient for a time series data observation from the 3^rd previous period if the original alpha value is set at .8?

A. The weight cannot be calculated since the data observation is not given.
B. The weight is zero since the alpha value is set relatively high.
C. .548
D. .0064

11.Given the data series below for variables Y (Monthly Inventory Balance) and X (Monthly Sales) are they significantly correlated at the 95% confidence level and how can you tell?

Ending Inv. Bal. Y		Monthly Sales X
1544		5053
1913		5052
2028		7507
1178		2887
1554		3880
1910		4454
1208		3855
2467		8824
2101		5716

A. Yes. The correlation coefficient is .873 that is greater than .05.
B. Yes. The correlation p-value is .002 which is less than .05.
C. No. The correlation coefficient is above the p-value.
D.No. The correlation p-value is greater than the 95% confidence level.

12. You have forecast the sales for your company for the last 12 months and the forecast residuals are shown below. Are these residuals to be considered random?
Residuals

-24

-348

-892

-62

-378

-489

-342

490

578

198

A. Yes, since the residuals randomly vary in magnitude.
B. Yes since the residuals are positive and negative and vary in magnitude.
C. No, since the residuals are stationary and vary in magnitude.
D. No, since the residuals indicate positive slope.

13. Given the residuals in problem 12 above what is the RMSE for the forecast?

A. -101.0
B. 411.8
C. 169603
D. 220.1

14. Which form of exponential smoothing can result in a na

Refer to the example of the Cobb-Douglas production function discussed in class, where the GDP (Y), Total Employment (X2 ), and Stock of Fixed Capital (X3 ), are measured for the output, labor input and capital input, respectively. The following model is fitted and the R output is below. The data are yearly data from 1955 to 1974.

lnY = B1 +B2lnX2 + B3lnX3 + u

a. Interpret the coefficient estimate of B3 specific to this problem.

b. Find the rate of change of Y with respect to the change of X2 , at Y=218506.3 and X2=11152.6, and X3 = 340072.45.

c. Find the estimate of the return to scale parameter. (3 pts.)

d. Is there strong evidence that the GDP is inelastic relative to the stocks of fixed capital? Justify your answer at a=0.05.

e. Test if there is a positive (first order) autocorrelation in the error term using a=0.05 . Make sure to state the null and alternative hypotheses, to provide the values of the test statistic and the critical value(s), and to draw a conclusion. (6 pts.)

Regression output for Problem 3.

> fit<-lm(log(y) ~ log(x2) + log(x3))

> summary(fit)

Call:

lm(formula = log(y) ~ log(x2) + log(x3))

Residuals:

Min 1Q Median 3Q Max

-0.057142 -0.016250 -0.005793 0.023512 0.038739

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.65233 0.60619 -2.726 0.0144 *

log(x2) 0.33972 0.18569 1.830 0.0849 .

log(x3) 0.84600 0.09335 9.063 6.42e-08 ***

Signif. codes: 0 0.001 0.01 0.05 0.1 1

Residual standard error: 0.02829 on 17 degrees of freedom

Multiple R-squared: 0.9951, Adjusted R-squared: 0.9945

F-statistic: 1719 on 2 and 17 DF, p-value: < 2.2e-16

Durbin-Watson Statistic = 0.425618

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EC260 Lecture Notes - Lecture 3: Ordinary Least Squares, Null Hypothesis, F Communications

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