1 Omitted Variable Bias: Part I. 2 Omitted Variable Bias: Part II. The Baseline: SLR.1-4 hold, and our estimates are unbiased

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1 Introductory Appled Econometrcs EEP/IAS 118 Sprng 2014 Andrew Crane-Droesch Secton #5 Feb Omtted Varable Bas: Part I Remember that a key assumpton needed to get an unbased estmate of β 1 n the smple lnear regresson s that E[u x] 0. If ths assumpton does not hold then we can t expect our estmate ˆβ 1 to be close to the true value β 1. We call ths problem omtted varable bas. That s, due to us not ncludng a key varable n the model, we have that E[ ˆβ 1 ] β 1. The motvaton of multple regresson s therefore to take ths key varable out of the error term by ncludng t n our estmaton. 2 Omtted Varable Bas: Part II The formula for omtted varable bas can be a lttle confusng, so to start we ll go through a few thngs much more slowly. Remember those SLR1-5 assumptons we talked about last tme? Prof. Buck stated n lecture that f SLR1-4 hold for a gven model, then our estmates of the ˆβ wll be unbased. Frst we re gong to take a closer look at what s gong wrong once we start thnkng about omtted varables. SLR4 fals because of an omtted varable: E[u X] 0 The Baselne: SLR.1-4 hold, and our estmates are unbased Populaton Model: Sample Regresson: What s the OLS formula for ˆβ 1? y β 0 + β 1 x + u ŷ ˆβ 0 + ˆβ 1 x ˆβ 1 Cov(x, y ) V ar(x ) (x x)(y ȳ) (x x) 2 (x x)y (x (See Appendx to these notes) x)x We can use what we know about the populaton model, plug y β 0 + β 1 x + u nto our formula for ˆβ 1 and smplfy: ˆβ 1 (x x)(β 0 + β 1 x + u ) (x x)x β 0 (x x) + β 1 (x x)x + (x x)u (x x)x β 1 + (x x)u (x x)x

2 Now, remember that ˆβ 1 s a random varable, so that t has an expected value: [ ] [ ] E ˆβ1 E β 1 + (x x)u (x x)x [ β 1 + E ] (x x)u (x β 1 x)x Aha! So under assumptons SLR.1-4, on average our estmates of ˆβ 1 wll be equal to the true populaton parameter β 1 that we were after the whole tme. 2

3 Realty Check: SLR.4 fals, E [u X] 0, and our estmates are based Populaton Model: Sample Regresson: What s the OLS formula for ˆα 1? ˆα 1 Cov(x, y ) V ar(x ) (x x)(y ȳ) (x x) 2 (x x)y (x x)x We can use what we know about the populaton model, plug y nto our formula for ˆα 1 and smplfy: ˆα 1 (x x)(β 0 + β 1 x + β 2 z + u ) (x x)x β 0 (x x) + β 1 (x x)x + β 2 (x x)z + (x x)u (x x)x β 1 + β 2 (x x)z (x + (x x)u x)x (x x)x There s an extra term! The second term β 2 (x x)z (x x)x s a result of our omsson of the varable z that affects y. When SLR.1-4 hold, on average our regresson estmates wll be close to the true parameters. But here, SLR.1-4 do not hold! If we take the expectaton of ˆα 1 : ] E [ˆα 1 ] E [β 1 + β 2 (x x)z (x + (x x)u x)x (x x)x [ ] [ ] β 1 + β 2 E (x x)z (x + E (x x)u x)x (x x)x β 1 + β 2 ρ 1 If E [ˆα 1 ] β 1 then we say ˆα 1 s based. What ths means s that on average, our regresson estmate s gong to mss the true populaton parameter by. 3 Example: OVB n Acton In ths secton, I use the wage data (WAGE1.dta) from your textbook to demonstrate the evls of omtted varable bas and show you that the OVB formula works. Let s pretend (!) that ths sample of 500 3

4 people s our whole populaton of nterest, so that when we run our regressons, we are actually revealng the true parameters nstead of just estmates. We re nterested n the relatonshp between wages and gender, and our omtted varable wll be tenure (how long the person has been at hs/her job). Suppose our populaton model s: log(wage) β 0 + β 1 female + β 2 tenure + u (1) Frst let s look at the correlatons between our varables and see f we can t predct how omttng tenure wll bas ˆβ 1 :. corr lwage female tenure lwage female tenure lwage female tenure If we ran the regresson:...then the nformaton above tells us that α 1 from runnng regressons (1) and (2): log(wage) α 0 + α 1 female + e (2) β 1. Let s see f we were rght. Below s the Stata output. reg lwage female tenure Source SS df MS Number of obs F( 2, 523) Model Prob > F Resdual R-squared Adj R-squared Total Root MSE lwage Coef. Std. Err. t P> t [95% Conf. Interval] female tenure _cons reg lwage female Source SS df MS Number of obs F( 1, 524) Model Prob > F Resdual R-squared Adj R-squared Total Root MSE.4935 lwage Coef. Std. Err. t P> t [95% Conf. Interval] female _cons Just to clarfy, we know that β 1 and α 1. Ths means that our BIAS s equal to: 4

5 β 1 + β 2 δ 1 There s one more parameter mssng from our OVB formula. What regresson do we have to run to fnd ts value? tenure ρ 0 + ρ 1 female + v (3) The Stata output from ths regresson s below:. reg tenure female Source SS df MS Number of obs F( 1, 524) Model Prob > F Resdual R-squared Adj R-squared Total Root MSE tenure Coef. Std. Err. t P> t [95% Conf. Interval] female _cons Just to clarfy, our ρ Now we can plug all of our parameters nto the bas formula to check that t n fact gves us the bas from leavng out tenure from our wage regresson: α 1 E[ˆα 1 ] ( )( ) OVB Intuton For further ntuton on omtted varable bas, I lke to thnk of an archer. When our MLR1-4 hold, the archer s amng the arrow drectly at the center of the target f he/she msses, t s due to random fluctuatons n the ar that push the arrow around, or maybe mperfectons n the arrow that send t a lttle off course. When MLR1-4 do not all hold, lke when we have an omtted varable, the archer s no longer amng at the center of the target. There are stll puffs of ar and feather mperfectons that send the arrow off course, but the course wasn t even the rght one to begn wth! The arrow (whch you should thnk of as our ˆβ) msses the center of the target (whch you should thnk of as our true β) systematcally. To demonstrate ths, I dd the followng: ˆ Take a random sample of 150 people out of the 500 that are n WAGE1.dta ˆ Estmate ˆβ 1 usng OLS, controllng for tenure wth these 150 people. ˆ Estmate ˆα 1 usng OLS (NOT controllng for tenure) wth these 150 people. ˆ Repeat 6000 tmes. At the end of all of the above, I end up wth 6000 based and 6000 unbased estmates of ˆβ 1. I plotted the kernel densty of the based estmates alongsde that of the unbased estmates. You can see how the based dstrbuton s shfted to the left ndcatng a downward bas! 5

6 Fgure 1. Kernel denstes for based and unbased estmates. Densty effect of female on ln(wage) alphahat_1 betahat_1 Take home practce problem: How to sgn the bas Traffc fataltes and prmary seatbelt laws. Usng data from Anderson (2008) for 49 US states, we can examne how prmary seatbelt laws (an offcer can pull you over just for not wearng your seatbelt) mpact annual traffc fataltes. From the paper, I have data on the number of traffc fataltes n 2000, whether or not the state had a pmary seatbelt law n place, and the total populaton of the state. In 2000, just 35% of the 49 states had prmary seatbelt laws (the rest had what s called a secondary seatbelt law). Suppose we run the followng regresson: fataltes ˆβ 0 + ˆβ 1 pop + ˆβ 2 prmary 1. Thnk of another varable or factor that you thnk affects traffc fataltes: 2. Is ths factor postvely or negatvely correlated wth f ataltes? + or 3. Is ths factor postvely or negatvely correlated wth prmary? + or 4. Omttng ths factor from our regresson wll bas ˆβ 1 : UPWARD or DOWNWARD Here are my results: f ataltes pop prmary Whoa! Accordng to our estmates, predcted fataltes ncrease wth the mplementaton of a prmary seatbelt law. Behavoral explanatons asde 1, omtted varables are the lkely culprts here. What are some varables that would nduce an upward bas n ˆβ 2? I thought that weather mght play a role n ths puzzle. States wth more dangerous weather wll have more traffc fataltes and are also more lkely to have a prmary seatbelt law: f ataltes pop prmary precp snow 1 By ths I mean arguments lke, addng safety requrements results n people behavng more recklessly. Whle often vald even n ths partcular case we re gong to keep t smple n ths dscusson. 6

7 (Clearly, even ths specfcaton wth controls for weather has some ssues: an addtonal nch of snow per year decreases predcted fataltes by lves?) 5 Confdence Intervals The smulaton that was shown n secton demonstrates somethng pretty profound: even after desgnng a random sample, collectng the data, fgurng out the populaton model, and runnng regressons, there s stll a chance your estmates are very far from those of the populaton. Each random sample yelds a dfferent estmate; f you have 100 random samples, you have 100 dfferent values of ˆβ 1. What can you do wth them? Confdence ntervals use the randomness of our sample estmates to say somethng useful about where the true populaton parameter actually s. You can thnk of confdence ntervals n two dfferent ways: 1. We can thnk of a confdence nterval as a bound for how wrong our sample estmate s. For example, f a poltcal poll fnds that a proposton wll receve 53.2% of the vote, we come to very dfferent conclusons f the margn of error s.5% or 5%. 2. Alternatvely, we can thnk of a confdence nterval as a measure of where the true, populaton value s lkely to be. (The wordng here s a lttle msleadng, as you ll see n a bt.) For example, f the true average wage for US laborers s $7, then t s unlkely that we d fnd a confdence nterval from our sample lke [10,14]. The bascs We can thnk of a sample mean, x the same way we thnk about our ˆβs: these are both. We know even more about x from the Central Lmt Theorem: For a random sample of a varable {x 1,..., x N }, the Central Lmt Theorem tells us that for very large samples (large N), the sample average x N(µ, σ 2 x). What ths means: f I took 10,000 dfferent random samples of laborers n the US and recorded ther wage, I would end up wth 10,000 dfferent sample means { x 1, x 2,..., x 10,000 }. If I plotted a hstogram of all of these sample means, t would look very much lke a normal dstrbuton and the center of the dstrbuton would be very close to the true average wage, µ, n the US. Because t s easy to get confused when we re talkng about a random varable X and another random varable x, whch s the sample mean of X, here s a table to keep thngs straght: Populaton Sample Mean of X: µ X Sample Mean: x 1 n x, and E[ x] µ X Varance of x: V ar(x) or σx 2 Sample Varance of x: s 2 1 n 1 (x x) 2, and E[s 2 ] σx 2 Varance of x: V ar( x) or σ 2 x Sample Varance of x: s 2 x s2 n, and E [ s 2 x] σ 2 x Normal dstrbutons are trcky to work wth, and t s easer to standardze normally dstrbuted varables so that they have a mean of 0 and a varance of 1. Remember our formula to fnd the expected value and varance of a transformed varable... If v s normally dstrbuted wth expected value E[v] and V ar(v) σv: 2 [ ] v E[v] E σ v 7

8 ( ) v E[v] V ar σ v Snce we re nterested n the dstrbuton of x (whch s normal), we can standardze t just lke above so x µ that: N(0, 1) σ x Now we can use what we know about the dstrbuton of standard normal varables to help us say somethng meanngful about what the true populaton mean, µ X mght be: ˆ We know that for any standard normal varable v, P r( 1.96 < v < 1.96) 95% ˆ We know that x µ X σ x s standard normal But we re not really nterested n the varable x µ X. The whole pont of ths s to learn more about µ σ x X! So we need to do some manpulaton of ths to solate µ X : A standard normal dstrbuton looks lke ths: If we draw a number z from a standard normal dstrbuton, then we know P r(1.96 < z < 1.96) Above we argued that x µ σ x N(0, 1) whch means that the P r( 1.96 < x µ σ x < 1.96) Just lke n lecture, we can rearrange terms to see that P r( x 1.96σ x < µ < x σ x ) How to nterpret a confdence nterval The most mportant thng to remember about a confdence nterval s that the what s random, not the. To make another metaphor out of an archac sport, I lke to thnk of confdence ntervals n the context of a game of horseshoes: s When to use the Student t dstrbuton: As you could guess from the table, n practce we do not know what σ x s, and we have to estmate t usng sample data wth: s 2 x, whch we call the standard error. Usng the standard error (whch s tself a random varable) changes the dstrbuton of the sample mean a lttle bt, and we have to use a Student s t dstrbuton nstead of a Normal dstrbuton: x µ s 2 n t n 1 The formula for a W% confdence nterval s: [ ( ) ( )] s s CI W x c W, x + c W n n Where c W s found by lookng at the t-table for n 1 degrees of freedom. 8

9 Step 1. Determne the confdence level. If we want to be 95% confdent that our nterval covers the true populaton parameter, then our confdence level s Pretty straght forward. Step 2. Compute your estmates of x and s. Step 3. Fnd c from the t-table. The value of c wll depend on both the sample sze (n) and the confdence level (always use 2-Taled for confdence ntervals): ˆ If our confdence level s 80% wth a sample sze of 10: c 80 ˆ If our confdence level s 95%, wth a sample sze of 1000: c 95 Step 4. Plug everythng nto the formula and nterpret. Plug our values of x, s and c nto the gven formula for a confdence nterval. nterpretaton: The trckest step s the ˆ Where does the randomness n a confdence nterval come from? ˆ So when we construct a confdence nterval, we nterpret t by sayng: Example I took a random sample of 121 UCB students heghts n nches, and found that x 65 and s 2 4. Followng the 4 steps above, I can fnd the 95% confdence nterval for x: 1. Confdence level was gven: 95% 2. x 65 and s 2 4 were gven. 3. From the t-table, c The 95% confdence nterval s [ , ] [64.64, 65.36]. Ths nterval has a 95% chance of coverng the true average heght of the populaton. Practce You have a random sample of housng prces n the Bay Area. After loadng the data nto Stata, you look at summary statstcs for the prces you observed: Varable Obs Mean Std. Dev. Mn Max prce Fnd a 99% confdence nterval for the true average housng prce: 9

10 6 Appendx Two facts used n the dscusson of omtted varable bas: (x x)(y ȳ) (x x)y (x x)ȳ (x x)y ȳ (x x) (x x)y ȳ(0) (x x)y Now replace every y n what s above to an x, and every ȳ to an x, and you can see that the same steps show (x x)2 (x x)x. 10