硕士计量1-08-Functional Forms-2

Adding More Regressors to Reduce the Error Variance

ˆ Given Var j

SST j 1 R
ˆ2
2 j

What is the effect of adding more variables ^ to our model on the Var(j)? It will reduce the error variance, because ˆ ˆ 2 ui2 / n k 1 SSR / n k 1 ^ And from it, it might reduce Var(j) if the added x’s are not correlated with xj
Which one is a better model? Can we use R2 to chose between the two? 2 R can be used to determine which is a better model

Choice of Models

Based on adjusted R2, the polynomial rather then the log model is a better model

Selection of Regressors (cont.)
On the other hand, if our purpose is to see the effect of lot size, number of rooms on the housing price, including the assessed value makes no sense, for it takes away the impact of lot size and number of rooms on the housing price Lets compare the two models in Eviews
2 R
Adjusted R2
Adding more variable to a model will not reduce R2 To compare two different models with different set of regressors (independent variables), the R2 adjusted by degree of R 2 gives us a better freedom denoted as means to compare the two It penalize a model with more regressors
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Sales 20000 25000
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35000
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RDIntens
8 6 4 2 0 0 2 4 Log(Sales) 6 8 10
Selection of Regressors
What should we include or exclude in our model depends on what the purpose of our study is For example, in the housing price model, if our purpose is to see if the assessed value of the house for tax purpose is rational, i.e. the assessed value reflect the market value of the house, then showing the inclusion of other factors made no difference completes the test

Standard Errors of the Intervals

First, suppose that we want an estimate of
The Mean Interval is a prediction based on the variability of the coefficient estimates only The Prediction Interval is a prediction taken also account of the variability of the error term Standard errors of the two are different

Two Types of Confidence Interval Predictions

There are two types of Confidence Interval predictions
Mean
Interval (an interval about the mean) Prediction Interval (an interval about a particular value)
R-squared 0.061329 Adjusted R-squared 0.03004
Mean dependent var 3.265625 S.D. dependent var 1.874079
Lets see why from the following graphs
10 9 8 7 6 5 4 3 2 1 0 0

Example: Choosing Between Two Models

See if we can use RDCHEM dataset to choose between:
rdintens= 0+1log(sales)+u rdintens= 0+1sales+2sales2+ u vs.
计量经济学（研究生）
Class 9 Functional Forms II
More on interaction terms Polynomial (Quadratic) Adjusted R2 Prediction Intervals
Adjusted Model
SSR / (n k 1) R 1 or TSS / (n 1)
2
1 (1 R 2 )(n 1) / (n k 1)
Example: Comparing Different Functional Forms with Adjusted R2

When choosing between two functional forms, for example:

Confidence Interval Predictions of Ŷ
Confidence Interval Predictions of Ŷ
Sometime we are interested in more than just the expected value of Y: ˆ ˆ ˆ ˆ ˆ E(Y ) Y 0 1 X1 2 X 2 ... k X k We might want to know with probability (1-) that an interval around E(Y) contains the true value of the dependent variable Just like we can compute confidence interval estimate of the coefficients, we can also compute confidence interval estimate of the predicted dependent value

If our purpose is to see the effect of lot size, rooms, and house size on the housing price, should we have included the asses value in the model, even if adding it will increase the Adjusted R-squre to .77?
y = 0+1log(x)+u y = 0+1x+ 2x2 +u vs.
Neither is F –test nor is t –test will help us much in choosing between two non-nested models Adjusted R2 comes to the rescue
Dependent Variable: Log(PRICE) No, that will have defeated our purpose. Method: Least Squares The assessed value has taken all these Sample: 1 88 factors into account, so we will not be able to gage the effects separately. Included observations: 88 Variable Coefficient Std. Error t-Statistic Prob. C -1.297041 0.651284 -1.991514 0.0497 Log(SQRFT) 0.700232 0.092865 7.540304 0.0000 BDRMS 0.036958 0.027531 1.342411 0.1831 Log(LOTSIZE) 0.167967 0.038281 4.387717 0.0000 R-squared 0.642965 Mean dependent var 5.633180 Adjusted R-squared 0.630214 S.D. dependent var 0.303573 S.E. of regression 0.184603 Akaike info criterion -0.496833 Sum squared resid 2.862563 Schwarz criterion -0.384227 Log likelihood 25.86066 F-statistic 50.42372 Durbin-Watson stat 2.088996 Prob(F-statistic) 0.000000