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# midterm 1 study guide Econ 421

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This 4 page Study Guide was uploaded by Theo Friedman on Friday April 17, 2015. The Study Guide belongs to Econ 421 at University of Oregon taught by Caroline Weber in Spring 2015. Since its upload, it has received 394 views. For similar materials see Intro Econometrics in Economcs at University of Oregon.

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Date Created: 04/17/15

Econ 421 study guide B population variable bsample variable U population error esample error 0 b s are random if sample changes b s change 0 with B never get the true number but on an average you will obtain the truth 0 sample approaches infinity you will be closer amp closer to the truth OLS assumptions 1 relation is linear with an added variable 2 X s Jointly random draws of yi xi from the population relationship X s have variation X is exogenous X is measured without error X causes Y but Y does not cause X lfthere is more than one X then it is multiple regression 3 Properties of the residual a Eui 0 zero mean b Eu 2 i o 2 i o 2 homoskedastic variance c Eui uj Eui Euj 0 Vi 6 j no spatial correlation in error terms d ui is normally distributed Standard error 0 Want them to be small 0 SEb1 sqrt squot2 xi xquot 2 9995 h Tests 0 Ttests are used to test a single coefficient 2 sided uses 1 sided uses gt or lt We fail to reject the null We can never accept the null tb1 B O SEb1 pvalue the probability under the assumption that H0 is true pvalue is smaller than the significance level chosen reject the null hypothesis there is also a critical T value anything greater than this we reject Ftest used when there is two jointly test for example B2 B3 or B2 B3 O F RSSr RSSuQ RSSrnk R2 Coefficient of determination Measures how well your model fits the data T 88 E88 R88 R2 ESST 88 1 RSST 88 R2 is bounded by O and 1 What minimizes RSS will maximize R2 R2 will never decrease and usually increases as you add variables to the model Adjusted R2 has been proposed to address this problem R2 1 RSSn k 1 T SSn 1 Coefficients 1 Continuous GP Ai BO B1Hi B2logSATi B3UP P ERi ui a A one unA one unit change in Xi changes Yi by B1 units b A 1 change in Xi changes Yi by B21OO units c Moving Xi from zero to one changes Yi by B3 2 Log a logGP Ai BO B1Hi B2logSATi B3UP P ERi ui b A one unit change in Xi changes Yi by 100B1 percent c A 1 change in Xi changes Yi by B2 percent d Moving Xi from zero to one changes Yi by 100B3 percent 3 Binary a HGP Ai BO B1Hi B2logSATi B3UP P ERi ui b one unit change in Xi changes the likelihood that Yi 1 by 100B1 percentage points c A 1 change in Xi changes the likelihood that Yi 1 by B2 percentage points d Xi from zero to one changes the like hood Yi 1 by 100B3 percentage points Other stuff 1 We call the estimate of the effect of the policy a treatment effect 2 GP Ai BO B1Hi B2UP P ERi B3Hi UP P ERi B4SATi ui a The intercept for underclassmen B0 and for upperclassmen BO B2 b The slope of Hi for underclassmen B1 and for upperclassmen B1 B3 3 Transformations You can have nonlinear relationships between Y and X but the coefficients must remain linear 4 Omitting a relevant variable that is correlated with an included variable 5 Including an irrelevant variable Heteroskedasticity o It does not bias the estimates 0 proof for bi start with b1 yi yxi x xi xquot2 0 end with E xi xui xi xquot2 O 0 Hence Eb1 B1 ie it is no longer BLUE it is inefficient OLS standard errors are inconsistent o t and F tests based upon the OLS variance estimates could be wrong HeteroskedasticRobust Standard Errors provides consistent estimates under both homoskedasticity and heteroskedasticity varb1 xi xquot2 equot2 xi xquot4 do not correct heteroskedasticity are not adjusted versions of OLS standard errors if error term is homoskedastic there is an efficiency gain from imposing this assumption Testing for Heteroskedasticity 1 BreuschPagan test a If oquot2 i is correlated with the x s we have heteroskedasticity estimates biased b Estimate the regression model c Calculate the residuals ei d Estimate the regression e Conduct an LM test where the null hypothesis is a1 ak O 2 White test a test for all types of relations between var ei amp X s b same as breusch pagan but with tons of variables White test will be more powerful against certain alternatives than the BP test but in modest samples using large number of degrees of freedom reduces power of test Other fixes o adopting a flexible specification is a strategy to reduce importance of heteroskedasticity o GLS The error term ui oi is homoskedastic because the variance is E quot ui oiquot2 1oquot2i Eu2 i 1 oquot2 i oquot2 i 1 0 FGLS If we don t know the form of oquot2 we may be able to estimate it o efficient but you have to accurately estimate heteroskedasticity Asymptotic Properties of OLS 0 consider asymptotic properties ie what happens when n gt such as consistency o plim is the probability limit that is it states what the variables converges to in probability as n gt oo 0 Rules for plim a plimXy plimXplimy b plimx y plimX plimy c plima a where a is a constant d plim X y plimX plimy e plimfX fpimx O 0000 Proof for consistence 0 start with b1 yi yxi x xi xquot2 o endwith B1 O oquot2xB1 Unbiasedness gt consistency but consistency 6gt unbiasedness Sometimes ui not normally distributed This is fine if performing our t or Ftest when n is large and ui is distributed asymptotically normal Measurement error Measurement Error in the Covariates we have the true population regression yi BO B1zi vi we don t observe zi Instead we observe xi where xi zi ooi 0 because 1 unable to measure 2 perfectly or 2 x is an imperfect proxy for 2 Plug equation for zi into the original equation yi BO B1xi ui Why is this inconsistent plimb1 B1 covx u varx combining the numerator amp denominator yields plimb1 B1 0quot2 2 0quot2 2 oquot2 00 We can see from this equation that estimates will be inconsistent biased towards zero Measurement error in the dependent variable does not matter as much

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