计量经济学课件
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• The basic framework will be the same for regressions. We estimate, test hypotheses about and construct confidence intervals for population parameters.
• We obtain the OLS estimator
������1 =
������ ������=1 ������������ −������ ������������ −������ ������ ������������ −������ 2 ������=1
������0 = ������ − ������1 ������
������ ������=1
������������ − ������0 − ������1 ������������
2
• We can solve this problem using calculus. The first order conditions are
������ ������=1
fitted value ������������ ≡ ������0 + ������1 ������������ , residual ������������ ≡ ������������ − ������������
• We choose ������0 and ������1 so that they minimize the sum of squared residuals: ������0 , ������1 argmin = ������0 , ������1
Number of obs F( 1, 524) Prob > F R-squared Adj R-squared Root MSE
= = = = = =
526 103.36 0.0000 0.1648 0.1632 3.3784
-----------------------------------------------------------------------------wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------educ | .5413593 .053248 10.17 0.000 .4367534 .6459651 _cons | -.9048516 .6849678 -1.32 0.187 -2.250472 .4407687 ------------------------------------------------------------------------------
• What is u? The error term or disturbance u represents
factors other than x that affect y. – We assume that u has zero conditional mean: ������ ������ ������ = 0 = ������ ������ . If this condition is satisfied, then we don’t have to control for the other factors.
20
average hourly earnings
Fitted (estimated) regression line: ������������������������ = −0.90 + 0.54 × ������������������������
• STATA output wage = ������0 + ������1 educ + ������
������ ������=1
������������ ������������ = 0
which implies that the OLS residual has zero sample covariance with x.
0
0
5
10
15
20
25
5
10 years of education
15 Fitted values
������������ − ������0 − ������1 ������������ = 0
and
������ ������=1
������������ ������������ − ������0 − ������1 ������������ = 0
• By solving the first order conditions for ������0 and ������1 ,
• Return to education wage = ������0 + ������1 educ + ������
– wage: dollars per hour, educ: years of education – ������1 measures the change in hourly wages given another year of education, holding all other factors fixed. – These factors include labor force experience, innate ability, tenure with current employer, work ethic, and so on.
Simple regression model
• Recap
– – – – Review of probability and statistics Estimation and hypothesis testing Confidence intervals As an example, we estimated a sample mean, tested a hypothesis about a population mean, and constructed confidence intervals for the population mean. – Test of the difference in means
. reg wage educ
Source | SS df MS -------------+-----------------------------Model | 1179.73204 1 1179.73204 Residual | 5980.68225 524 11.4135158 -------------+-----------------------------Total | 7160.41429 525 13.6388844
������ ≡
2
������������������ ������������������
=1
������������������ − ������������������
– Unitless and ranges between zero (no fit) and one (perfect fit) – ������2 = 0 means ������������������ = 0 – ������2 = 1 means ������������������ = ������������������ – 0 ≤ ������2 ≤ 1 with an intercept – Does low ������2 mean the OLS regression is useless?
• Measure of goodness-of-fit
– How well does the OLS regression line fit or explain the data? R-squared or ������������ measures the fraction of the sample variation in y that is explained by x.
– Estimation: how to use the data to obtain estimates of ������0 and ������1 ? – Hypothesis testing: how to test whether the slope is a particular value, e.g. zero? – Confidence intervals: how to construct a confidence interval for the slope?
������ = ������0 + ������1 ������ + ������
Ordinary least squares (OLS)
������������ = ������0 + ������1 ������������ + ������������ = ������������ + ������������
– Population – ������: dependent variable (explained variable, regressand) – ������: independent variable (explanatory variable, regressor) – ������0 : intercept – ������1 : slope. x has a linear effect on y as Δ������ = ������1 Δ������ if Δ������ = 0
������������ = ������������ + ������������
������ ������=1
������������Βιβλιοθήκη Baidu− ������
2
=
������ ������=1
������������ − ������
2
+
������ ������=1
������������
2
• Total sum of squares = explained SS + residual SS (SST = SSE + SSR)
- Sample mean minimizes ������ ������������ − ������ 2 ������=1 - Why minimize the sum of squared residuals?
• ������������ = ������������ − ������0 + ������1 ������������ is called the OLS residual. In particular, ������−1
• Population regression line (function)
������ ������ ������ = ������0 + ������1 ������ – ������1 : How much does y vary on average when x changes?
• We don’t know ������0 and ������1 and have to estimate them. Statistical inference for linear regression models is in general the same as for population means.
• We are interested in explaining y in terms of x or investigating how y varies as x changes. • Simple linear regression (SLR) model
������ = ������0 + ������1 ������ + ������