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Regression Printout - Econometrics - Lecture Notes, Study notes of Econometrics and Mathematical Economics

Regression Printout, Fitted values or the regression line, Regression equation, Residuals, Estimated or Regression sum of Square, Residuals Sum, R square, Standard Error are points you can learn about Econometric in this lecture.

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2011/2012

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Chapter 03
The Regression Printout
We have already learned how to run the regression in Microsoft Excel. When you run a regression, you
will see lot of output on screen. A sample output is given below;
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
ANOVA
SS
MS
F
Significance F
Regression
62.20
62.20
249.96
5.84E-17
Residual
8.21
0.25
Total
70.41
Standard Error
t Stat
P-value
Intercept
0.084
-0.638
0.528
X
0.067
15.810
0.000
The purpose of this lesson is to understand the numbers appearing on the screen, the algorithm to compute
them and what these numbers stand for.
Consider the regression equation: ๐‘ฆ๐‘–=๐›ผ+๐›ฝ๐‘ฅ๐‘–+๐œ€๐‘–
๏‚ง The first thing in regression analysis is to calculate the estimates of ๐›ผ ๐‘Ž๐‘›๐‘‘ ๐›ฝ. These estimates can
be calculated by the formula we have learned in lecture 2 or by the matrix approach.
๏‚ง The remaining output visible in this regression output is calculated from these two estimates and
the data set. The detail is as follows:
Fitted values or the regression line:
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Chapter 03

The Regression Printout

We have already learned how to run the regression in Microsoft Excel. When you run a regression, you will see lot of output on screen. A sample output is given below;

SUMMARY OUTPUT

Regression Statistics Multiple R (^) 0. R Square (^) 0. Adjusted R Square (^) 0. Standard Error (^) 0. Observations (^35) ANOVA df SS MS F Significance F Regression 1 62.20 62.20 249.96 5.84E- 17 Residual 33 8.21 0. Total 34 70.

Coefficients Standard Error t Stat P-value Intercept (^) - 0.054 0.084 - 0.638 0. X (^) 1.064 0.067 15.810 0.

The purpose of this lesson is to understand the numbers appearing on the screen, the algorithm to compute them and what these numbers stand for.

Consider the regression equation: ๐‘ฆ๐‘– = ๐›ผ + ๐›ฝ๐‘ฅ๐‘– + ๐œ€๐‘–

๏‚ง The first thing in regression analysis is to calculate the estimates of ๐›ผ ๐‘Ž๐‘›๐‘‘ ๐›ฝ. These estimates can be calculated by the formula we have learned in lecture 2 or by the matrix approach. ๏‚ง The remaining output visible in this regression output is calculated from these two estimates and the data set. The detail is as follows:

Fitted values or the regression line:

For any value of variable of X-variable, prediction of corresponding value of variable can be obtained by

following formula: ๐‘ฆ๏ฟฝ = ๐›ผ๏ฟฝ + ๐›ฝฬ‚๐‘ฅ

The vector ๐‘ฆ๏ฟฝ is the vector of estimates of Y for all values of variable X in the data set. These values formulate a straight line and represent the predictable part of variable Y.

Residuals:

The residuals are calculated as follows:

๐‘’๐‘– = ๐‘ฆ๐‘– โˆ’ ๐‘ฆ๏ฟฝ๐‘–

Estimated or Regression Sum of Square

This is given by the formula ๐ธ๐‘†๐‘† = โˆ‘^ ๐‘›๐‘–=1(๐‘ฆ๏ฟฝ๐‘– โˆ’ ๐‘ฆ๏ฟฝ)^2

ESS represents the part of variation in Y that is estimated by the regression, therefore it is called estimated sum of squares.

Residuals Sum of Squares

This is given by the formula ๐‘…๐‘†๐‘† = โˆ‘ ๐‘›๐‘–=1( ๐‘ฆ๐‘– โˆ’ ๐‘ฆ๏ฟฝ๐‘–)^2 = โˆ‘ ๐‘›๐‘–=1๐‘’๐‘–^2

RSS is sum of square of residuals which is difference between true values of Y and its estimates via regression equation. Therefore it represents the part of variation in Y that is estimated by the regression, therefore it is called estimated sum of squares.

Total Sum of Squares

The formula for Total Sum of square is: ๐‘‡๐‘†๐‘† = โˆ‘^ ๐‘›๐‘–=1(๐‘ฆ๐‘– โˆ’ ๐‘ฆ๏ฟฝ)^2

The formula indicates that TSS is measure of total variation in Y. It can be proven mathematically that TSS=ESS+RSS

R-square

The formula for R โ€“square is:

R-square=ESS/TSS

We know that ESS is the portion of variation in Y that is explained using the regression equation, whereas the TSS is measure of total variation in Y. Therefore R-square represent percentage of variation in Y that is explained by the regression.

Note that we have ESS=TSS-RSS

Therefore ๐‘… โˆ’ ๐‘ ๐‘ž๐‘ข๐‘Ž๐‘Ÿ๐‘’ = ๐‘‡๐‘†๐‘†โˆ’๐‘…๐‘†๐‘† ๐‘‡๐‘†๐‘† = 1^ โˆ’^

๐‘…๐‘†๐‘† ๐‘‡๐‘†๐‘†

R-square is called coefficient of determination. This tells us how much part of variation in the dependent variable can be explained by using the variable X. The square root of R-square is called the multiple R.

โˆ’(โˆ‘ ๐‘ฅ๐‘–)^2 + ๐‘› โˆ‘ ๐‘ฅ๐‘–^2

And

Therefore we got the regression equation

Further output is calculated as follows:

X Y ๐‘ฆ๏ฟฝ = ๐›ผ๏ฟฝ + ๐›ฝฬ‚ ๐‘ฅ ๐‘’๐‘–^ =^ ๐‘ฆ๐‘–^ โˆ’^ ๐‘ฆ๏ฟฝ๐‘–^ ๐‘ฆ๐‘–^ โˆ’^ ๐‘ฆ๏ฟฝ

๏‚ง ESS is sum of squares of the last column which is 22. ๏‚ง RSS is sum of squares of second last column which is 5.15. ๏‚ง TSS=RSS+ESS=22.85+5.15=28.

Therefore

๏‚ง R-square=ESS/TSS=22.85/28=0. ๏‚ง Multiple R =0.90 and

๏‚ง Standard Error ๐œŽ๏ฟฝ = ๏ฟฝ^ ๐‘…๐‘†๐‘† ๐‘›โˆ’๐‘˜ =^ ๏ฟฝ

  1. 15 4 = 1.

The Graph of above mentioned model is as follows:

Table 1 X Y 10 45 21 38 28 56 44 71 50 98 62 81

Exercise 1: For the data in Table 1, calculate the following statistics

  1. The estimated of parameters for regression equation ๐‘ฆ = ๐›ผ + ๐›ฝ๐‘ฅ + ๐œ€
  2. The standard error of regression ๐œŽ๏ฟฝ
  3. ESS, TTS and R-square

Exercise 2: Suppose RSS=10.51 and n=

  1. Calculate ๐œŽ๏ฟฝ
  2. If R-square=0.81, calculate ESS and TSS

Exercise 3: Suppose for some data set, estimated regression equation is ๐‘ฆ = 2.51 + 0.53๐‘ฅ + ๐‘’

  1. Predict the value of y when x=