REGRESSION ASSUMPTIONS Best Linear Unbiased Estimate BLUE If
REGRESSION ASSUMPTIONS
Best Linear Unbiased Estimate (BLUE) If the following assumptions are met: � The Model is � Complete Linear Additive Variables are � measured at an interval or ratio scale without error The regression error term is � unrelated to predictors normally distributed has an expected value of 0 errors are independent homoscedasticity In a system of interrelated equations the errors are unrelated to each other Characteristics of OLS if sample is probability sample � � Unbiased Efficient Consistent
The Three Desirable Characteristics � Unbiased: � E(b)=β � � On the average we are on target Efficient � � b is the sample β is the true, population coefficient Standard error will be minimum Consistent � As N increases the standard error decreases and closes in on the population value
Completeness . regress API 13 MEALS AVG_ED P_EL P_GATE EMER DMOB if AVG_ED>0 & AVG_ED<6, beta Source | SS df MS -------+-------------------Model | 65503313. 6 6 10917218. 9 Residual | 37321960. 3 10075 3704. 41293 -------+-----------------------------------Total | 102825274 10081 10199. 9081 Number of obs F( 6, 10075) Prob > F R-squared Adj R-squared Root MSE = 10082 = 2947. 08 = 0. 0000 = 0. 6370 = 0. 6368 = 60. 864 ------------------------------------------------------API 13 | Coef. Std. Err. t P>|t| Beta -------+-----------------------------------------------MEALS |. 1843877. 0394747 4. 67 0. 000. 0508435 AVG_ED | 92. 81476 1. 575453 58. 91 0. 000. 6976283 P_EL |. 6984374. 0469403 14. 88 0. 000. 1225343 P_GATE |. 8179836. 0666113 12. 28 0. 000. 0769699 EMER | -1. 095043. 1424199 -7. 69 0. 000 -. 046344 DMOB | 4. 715438. 0817277 57. 70 0. 000. 3746754 _cons | 52. 79082 8. 491632 6. 22 0. 000. ------------------------------------------------------ Meals . regress API 13 MEALS AVG_ED P_EL P_GATE EMER DMOB PCT_AA PCT_AI PCT_AS PCT_FI PCT_HI PCT_PI PCT_MR if AVG_ED>0 & AVG_ED<6, beta Source | SS df MS --------+----------------------------------Model | 67627352 13 5202104 Residual | 35197921. 9 10068 3496. 01926 -------+-----------------------------------Total | 102825274 10081 10199. 9081 Number of obs F( 13, 10068) Prob > F R-squared Adj R-squared Root MSE -------------------------------------------------------API 13 | Coef. Std. Err. t P>|t| Beta -------+-----------------------------------------------MEALS |. 370891. 0395857 9. 37 0. 000. 1022703 AVG_ED | 89. 51041 1. 851184 48. 35 0. 000. 6727917 P_EL |. 2773577. 0526058 5. 27 0. 000. 0486598 P_GATE |. 7084009. 0664352 10. 66 0. 000. 0666584 EMER | -. 7563048. 1396315 -5. 42 0. 000 -. 032008 DMOB | 4. 398746. 0817144 53. 83 0. 000. 349512 PCT_AA | -1. 096513. 0651923 -16. 82 0. 000 -. 1112841 PCT_AI | -1. 731408. 1560803 -11. 09 0. 000 -. 0718944 PCT_AS |. 5951273. 0585275 10. 17 0. 000. 0715228 PCT_FI |. 2598189. 1650952 1. 57 0. 116. 0099543 PCT_HI |. 0231088. 0445723 0. 52 0. 604. 0066676 PCT_PI | -2. 745531. 6295791 -4. 36 0. 000 -. 0274142 PCT_MR | -. 8061266. 1838885 -4. 38 0. 000 -. 0295927 _cons | 96. 52733 9. 305661 10. 37 0. 000. ------------------------------------------------------ = 10082 = 1488. 01 = 0. 0000 = 0. 6577 = 0. 6572 = 59. 127 Parents’ education
Diagnosis and Remedy � Diagnosis � Theoretical � Remedy � Including new variables
Linearity Violation of linearity An almost perfect relationship will appear as a weak one Almost all linear relations stop being linear at a certain point
Diagnosis & Remedy Diagnosis: � � � Visual scatter plots Comparing regression with continuous and dummied independent variable Remedy: � � Use dummies Y=a+b. X+e becomes Y=a+b 1 D 1+ …+bk-1 Dk-1+e where X is broken up into k dummies (Di) and k-1 is included. If the Rsquare of this equation is significantly higher than the R-square of the original that is a sign of nonlinearity. The pattern of the slopes (bi) will indicate the shape of the non-linearity. � � � Transform the variables through a non-linear transformation, therefore � � � � Y=a+b. X+e becomes Quadratic: Cubic: Kth degree polynomial: Y=a+b 1 X+b 2 X 2+e Y=a+b 1 X+b 2 X 2+b 3 X 3+e Y=a+b 1 X+…+bk. Xk+e Logarithmic: Exponential: Inverse: Y=a+b*log(X)+e or log(Y)=a+b. X+e or Y=ea+bx+e Y=a+b/X+e etc.
Example
Meaningless! Inflection point: -b 1/2*b 2 -(-3. 666183)/2*. 0181756=100. 85425 As you approach 100% the negative effect disappears
Additivity � � The assumption is that both X 1 and X 2 each, separately add to Y regardless of the value of the other. � � Y=a+b 1 X 1+b 2 X 2+e E. g. Inc=a+b 1 Education+b 2 Citizenship+e Imagine, that the effect of X 1 depends on X 2. � � If Citizen Inc=a+b*1 Education+e* If Not Citizen Inc=a+b**1 Education+e** where b*1 >b**1 � � You cannot simply add the two. If Citizenship is takes only two values, their effect is multiplicative: � Inc=a+b 1 Education*b 2 Citizenship+e There are many examples of the violation of additivity: � � � E. g. , the effect of previous knowledge (X 1) and effort (X 2) on grades (Y) The effect of race and skills on income (discrimination) The effect of paternal and maternal education on academic achievement
Diagnosis & Remedy � Diagnosis: � Try other functional forms and compare R-squares � Remedy: � Introducing the multiplicative term as a new variable so Yi=a+b 1 X 1+b 2 X 2+e becomes � Yi=a+b 1 X 1+b 2 X 2+b 3 Z+ e where Z=X 1*X 2 � Or transforming the equation into additive form � If Y=a*X 1 b 1*X 2 b 2*e then � log Y=log(a)+b 1 log(X 1)+b 2 log(X 2)+e so �
Example with one dummy variable Model Summary Model R R Square 1. 720(a). 519 a Predictors: (Constant), ESCHOOL, AVG_ED Does parents’ education matter more in elementary school or later? Adjusted R Square Std. Error of the Estimate. 519 70. 918 Coefficients(a) Model 1(Constant) AVG_ED ESCHOOL a Dependent Variable: API 13 Unstandardized Coefficients B Std. Error 510. 030 2. 738 87. 476. 930 54. 352 1. 424 Standardized Coefficients Beta. 649. 264 t Sig. 186. 250 94. 085 38. 179 . 000 Model Summary Model R R Square Adjusted R Square 1. 730(a). 533 a Predictors: (Constant), INTESXED, AVG_ED, ESCHOOL Std. Error of the Estimate 69. 867 Coefficients(a) Model 1(Constant) AVG_ED ESCHOOL AVG_ED*ESCHOOL(interaction) a Dependent Variable: API 13 Unstandardized Coefficients B Std. Error 454. 542 4. 151 107. 938 1. 481 145. 801 5. 386 -33. 145 1. 885 Standardized Coefficients Beta. 801. 707 -. 495 t Sig. 109. 497 72. 896 27. 073 -17. 587 . 000
Equations � Pred(API 13 i)= 454. 542+ 107. 938*AVG_EDi+ 145. 801*ESCHOOLi+(-33. 145)*AVG_EDi*ESCHOOLi � � IF ESCHOOL=1 i. e. school is an elementary school Pred(API 13 i)= 454. 542+ 107. 938*AVG_EDi+ 145. 801*1+(-33. 145)*AVG_EDi*1 = 454. 542+ 107. 938*AVG_EDi+ 145. 801+(-33. 145)*AVG_EDi = (454. 542 + 145. 801)+ (107. 938 -33. 145)*AVG_EDi = � 600. 343+74. 793*AVG_EDi � � IF ESCHOOL=0 i. e. school is not an elementary but a middle or high school Pred(API 13 i)= 454. 542+ 107. 938*AVG_EDi+ 145. 801*0+(-33. 145)*AVG_EDi*0 = � 454. 542+ 107. 938*AVG_EDi � The effect of parental education is larger after elementary school! Is this difference statistically significant? � � � Coefficients(a) Model 1(Constant) AVG_ED ESCHOOL AVG_ED*ESCHOOL(interaction) a Dependent Variable: API 13 Unstandardized Coefficients B Std. Error 454. 542 4. 151 107. 938 1. 481 145. 801 5. 386 -33. 145 1. 885 Standardized Coefficients Beta. 801. 707 -. 495 t Sig. 109. 497 72. 896 27. 073 -17. 587 . 000
Example with continuous variables
Proper Level of Measurement
Measurement Error Take Y=a+b. X+e where X is the real value and e is a random measurement error � � Suppose X*=X+e � Then Y=a+b’X*+e’ Y=a+b’(X+e)+e’=a+b’X+b’e+e’ � Y=a+b’X+E where E=b’e+e’ and b’=b � The slope (b) will not change but the error will increase as a result � Our R-square will be smaller Our standard errors will be larger t-values smaller significance smaller Suppose X#=X+c. W+e where W is a systematic measurement error c is a weight � Then Y=a+b’X#+e’ Y=a+b’(X+c. W+e)+e’=a+b’X+b’c. W+E � b’=b iff rwx=0 or rwy=0 otherwise b’≠b which means that the slope will change together with the increase in the error. Apart from the problems stated above, that means that Our slope will be wrong
Diagnosis & Remedy � Diagnosis: � Look at the correlation of the measure with other measures of the same variable � Remedy: � Use multiple indicators and structural equation models (AMOS) � Confirmatory factor analysis � Better measures
Normally Distributed Error
Non-Normal Error Our calculations of statistical significance depends on this assumption � Statistical inference can be robust even when error is nonnormal � Diagnosis: � You can look at the distribution of the error. Because of the homoscedasticity assumption (see later) the error when summed up for each prediction should be also normal. (In principle, we have multiple observations for each prediction. ) � Remember! Our measured variables (Y and X) do not have to have a normal distribution! Only the error for each prediction. � � Remedy: � Any non-linear transformation will change the shape of the distribution of the error
Example: Count Data N childs NUMBER OF CHILDREN 1751 Minimum Maximum 0 8 Mean 1. 89 DEPENDENT VARIABLE Underdispersion : Mean/Std. Dev. >1 Overdispersion : Mean/Std. Dev. <1 As Mean >Std. Deviation we have a case of a (small) underdispersion Std. Deviation 1. 665
Poisson and Negative Binomial Regressions Poisson assumes Mean=Std. Dev (No over- or underdispersion) Negative Binomial does not make this assumption Log of expected counts is now the unit of the dependent variable
Error Has a Non-Zero Mean The solid line gives a negative The dotted line a positive mean This can happen when we have some selection problem Diagnosis: 1. 000. 998. 996. 994. 992. 990 Z . 988. 986 Rsq = 0. 6211 200 400 600 800 1000 1200 X Visual scatter plot will not help unless we know in advance somehow the true regression line Remedy: If it is a selection problem try to address it.
Non-independent errors Example 1: Suppose you take a survey of 10 people but you interview everyone 10 times. Now your N=1000 but your errors are not independent. For the same person you will have similar errors � � Example 2: Suppose you take 10 countries and you observe them in 10 different time period Now your N=1000 but your errors are not independent. For the same country you will have similar errors � � � Example 3: Suppose you take 100 countries and you observe them only once. Now your N=100. But countries that are next to each other are often similar (same geography and climate, similar history, cooperation etc. ). If your model underpredicts Denmark, it is likely to underpredict Sweden as well. � Example 4: Suppose you take 100 people but they are all couples, so what you really have is 50 couples. Husband wife tend to be similar. If your model underestimates one chances are it does the same for the other. Spouses have similar errors. Statistical inference assumes that each case is independent of the other and in the two examples above it is not the case. In fact, your N < 100. � � � This biases your standard error because the formula is “tricked into believing” that you have a larger sample than you actually have and larger samples give smaller standard errors and better statistical significance. This may also bias your estimates of the intercept and the slope. Non-linearity is a special case of correlated errors.
Diagnosis & Remedy � � � It is called autocorrelation because the correlation is between cases and not variables, although autocorrelations often can be traced to certain variables such as common geographic location or same country or person or family. Diagnosis � Visual, scatterplot � Checking groups of cases that are theoretically suspect � Certain forms of serial or spatial autocorrelations can be diagnosed by calculating certain statistics (e. g. , Durbin-Watson test) Remedy: � You can include new variables in the equation � E. g. : for serial (temporal) correlation you can include the value of Y in t-1 as an independent variable � For spatial correlation we can often model the relationships by introducing an weight matrix
Heteroscedasticity � Homoscedasticity means equal variance Heteroscedasticity means unequal variance We assume that each prediction is not just on target on average but also that we make the same amount of error Heteroscedasticity results in biased standard errors and statistical significance � Diagnosis: � � Visual, scatter plot � Remedy: � Introducing a weight matrix (e. g. using 1/X)
Predictor Related to Error � � � Error represents all factors influencing Y that are not included in the regression equation If an omitted variable is related to X the assumption is violated. This is the same as the Completeness or Omitted Variable Problem Diagnosis: � The error will ALWAYS be uncorrelated with X, there is no way to establish the TRUE error � Theoretical � Remedy: � Adding new variables to the model
Correlated errors across interrelated equations � We sometimes estimate more than one regression. � Suppose Yt=a+b 1 Xt-1+b 2 Zt-1+e but � Xt=a’+b’ 1 Yt-1+b’ 2 Zt-1+e’ � e and e’ will be correlated � (whatever is omitted from both equations will show up in both e and e’ making them correlated) � This is also the case in sample selection models � S=a+b 1 X+b 2 Z+e S is whether one is selected into the sample � Y=a+b’ 1 X+b’ 2 Z+b’ 3 W+b’ 4 V+e’ Y is the outcome of interest e and e’ will be correlated (whatever is omitted from both equations will show up in both e and e’ making them correlated) � �
- Slides: 27
