in residuals.
Private investment has a significant negative effect on poverty rate. Every 1 billion dollars increase in private investment is associated with 0.0079 percentage point decrease in total poverty rate.
State-local expenditures has a significant negative effects on poverty rate. Every 1 billion dollars increase in the stat-local expenditure is associated with 0.027 percentage point drop in poverty rate.
Unemployment rate has a positive effect with poverty rate. Every 1 percentage point increase in unemployment rate is associated with 0.844 percentage point increase in poverty rate.
Violent crime rate has a negative effect on poverty rate. Every 1 per 100,000 crime increase is associated with 0.0222 percentage point decrease in poverty rate. GDP has a significant positive effect with poverty rate. Every 1 billion dollars increase in GDP is associated with 0.0052 percentage point increase in poverty rate.
Fixed investment, government expenditures and consumptions have insignificant effects on poverty rate. (3.c)Create the lag-poverty rate.
(3.d) Run a regression model with the total poverty rate as the response variable and Lag-poverty, private
investment, Fixed investment, Unemployment rate, Government expenditure, State and local expenditures, Consumption, GDP, and violent crime rate as input variables. Save Residuals and test the independence hypothesis of the residuals at the 5% significant level.
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Use the following program command:
proc autoreg data=Sasuser.Q2; GOVERNMENT_EXP STATE_LOCAL UNEMPLOYMENT_RATE GDP R_VIOLENT_CRIME /lagdep= LAGPOVERTYRATE; quit;
we can derive Durbin h statistics.
Durbin h 0.6154 Pr > h 0.2692
So the residuals are independent at the 5% significant level.
(3.e) Check the normality assumption of the residuals. Interpret the regression estimates.
model POVERTYRATE_TOTAL = LAGPOVERTYRATE CONSUMPTION PRIVATE_INVEST FIXED_INVEST
run;
All three normality tests fail to reject the null hypothesis of normality, i.e., the normality assumption is satisfied.
With the adjustment of lag-poverty rate in the model, we only find private investment still has a negative effect and unemployment has a positive effect, and fixed investment has positive effect. All other variables have insignificant effects individually.
(3.f) Test the hypothesis that the five variables of Government expenditure, State and local expenditures,
Consumption, GDP, and violent crime rate have no effects on violent crime when variables of Lag-poverty, private investment, Fixed investment, and Unemployment rate are included in the model at 5% level of significance.
Average explaining power by the variables under test=SS (Government expenditure, State and local expenditures,
Consumption, GDP, and violent crime rate/ Lag-poverty, private investment, Fixed investment, and Unemployment rate)/ 5 = (0.071 + 0.135 + 0.053 + 0.930 + 0.053 ) /5 =1.242/5=0.2484 Average variation of errors= SSE /(34)= 0.259 ;
F-statistics = 0.2484/0.259 = .9594 < 2.49362 (95 percentile of F5,34 is 2.49362) 也可直接用test语句得到F值和P值。
We fail to reject the null hypothesis. That is to say these five variables have no additional effect on violent crime at 5% significance level when the other 4 input variables are included in the model.
4. Let Y and X denote the height (ft) of the tree and the diameter (inches) at breast height respectively. The least
square estimate of the simple regression model relating Y to X is as follows:
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Y = 9.15 +.481 X + e (4.a) Interpret the parameter estimate of ????
When the diameter at breast height increases by 1 inch, the height of the tree increases by 0.481 feet on average.?
???b) Find SST, SSR and SSE and comment on the goodness of fit for the above model.
F=65.10 > F(1,34)(p-value<0.0001); The null hypothesis is rejected. X has significant effect on Y.
Root MSE = 1.67773 (ft). The average prediction error made by using the regression is 1.6777 feet.
Compare to the average height of the trees 17.908 feet, this is moderately acceptable (the forecast error is about 10%).
R-Square = 65.7%, 65.7% of the total variation (SST) is explained by the regression with X as the input variable.
Based on the above analysis, the model fits the data moderately well. (4.c) Consider the second order model:
Y??0??1X??2X2?e (4.2) Calculate the least square estimate.
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(4.d) Is it useful to have the second order term in the regression equation? (hint: test the hypothesis that ?????????
?
??.e) Let SSR(X) denote the SSR in the first order model and SSR(X,X) denote the SSR in the second order
model. Do you see improvement of the second order model in the comparison of SSR(X) and SSR(X,X2)? SSR(X)=183.245 from model (1)
SSR(X, X**2) = 215.941 from model(2).
Model (2) can explain more variation of Y. How much more? 215.941- 183.245=32.696. is this really significant? Let’s compare that with the average squared errors (2.815 in model (1)).
That’s about 10 times of the squared errors. In that sense, we would conclude the improvement is significant.
(4.f) Let SSR(X2| X) = SSR(X,X2)-SSR(X) denote the marginal contribution of the second order term in equation
(4.2). Comparing this marginal contribution with the MSE (mean sum of squared errors) of Model (4.2), do you see evidence of improvement with the second order model?
SSR(X**2 | X) == SS(X, X**2) – SS(X) = 32.696 comparing with the average squared error of model (2) (1.909). It is more than 16 times of the average squared errors. Yes, the improvement is significant.
(4.g) Plot the fitted line of the first order model and the second order model respectively. Which model fits the
data better graphically? Is your finding here consistent with results obtained in (4.f)?
Graphically, the quadratic model fits the data better, which is consistent with the findings in the previous
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regression results.
(4.h) Interpret the estimated results of Model (4.2). Is Model (4.2) an explanatory model or predictive model? or
can it be both ways?
Since model (4.2) fits the data well, it provides a better explaining power. We have not done any checking on the performance of prediction. It is not clear if this would be a good predictive model yet. To check if a model is a good predictive model, we should keep some data for checking purpose.
5.Does Ephemeral Products, Inc., practice sex discrimination in determining employee salaries?
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