Lecture 13. Use and Interpretation of Dummy Variables. Stop worrying for 1 lecture and learn to appreciate the uses that dummy variables can be put to
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1 Lecture 13. Use and Interpretation of Dummy Variables Stop worrying for 1 lecture and learn to appreciate the uses that dummy variables can be put to Using dummy variables to measure average differences Using dummy variables when more than 2 discrete categories Using dummy variables for policy analysis Using dummy variables to net out seasonality
2 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative
3 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way
4 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied
5 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied D = 0 if not
6 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied D = 0 if not Eg. Male/Female
7 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied D = 0 if not Eg. Male/Female so that the dummy variable Male would be coded 1 if male
8 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied D = 0 if not Eg. Male/Female so that the dummy variable Male would be coded 1 if male and 0 if female
9 Use and Interpretation of Dummy Variables Dummy variables where the variable takes only one of two values are useful tools in econometrics, since often interested in variables that are qualitative rather than quantitative In practice this means interested in variables that split the sample into two distinct groups in the following way D = 1 if the criterion is satisfied D = 0 if not Eg. Male/Female so that the dummy variable Male would be coded 1 if male and 0 if female (though could equally create another variable Female coded 1 if female and 0 if male)
10 Example: Suppose we are interested in the gender pay gap
11 Example: Suppose we are interested in the gender pay gap Model is LnW = b0 + b1age + b2male where Male = 1 or 0
12 Example: Suppose we are interested in the gender pay gap Model is LnW = b0 + b1age + b2male where Male = 1 or 0 For men therefore the predicted wage ^ ^ ^ LnW men = b0 + b1 Age + ^ b2*(1)
13 Example: Suppose we are interested in the gender pay gap Model is LnW = b0 + b1age + b2male where Male = 1 or 0 For men therefore the predicted wage ^ ^ ^ LnW men = b0 + b1 Age + ^ b2*(1) ^ ^ ^ = b 0 + b1 Age + b2
14 Example: Suppose we are interested in the gender pay gap Model is LnW = b0 + b1age + b2male where Male = 1 or 0 For men therefore the predicted wage ^ ^ ^ LnW men = b0 + b1 Age + ^ ^ ^ = b 0 + b1 Age + b2 ^ b2*(1) For women ^ ^ ^ ^ LnW women = b0 + b1 Age + b2*(0)
15 Example: Suppose we are interested in the gender pay gap Model is LnW = b0 + b1age + b2male where Male = 1 or 0 For men therefore the predicted wage ^ ^ ^ LnW men = b0 + b1 Age + ^ ^ ^ = b 0 + b1 Age + b2 ^ b2*(1) For women ^ ^ ^ ^ LnW women = b0 + b1 Age + b2*(0) ^ ^ = b0 + b1 Age
16 Remember that OLS predicts the mean or average value of the dependent variable (see lecture 2) Y ˆ = Y So in the case of a regression model with log wages as the dependent variable, LnW = b0 + b1age + b2male the average of the fitted values equals the average of log wages _ ^ _ Ln ( W) = LnW
17 Remember that OLS predicts the mean or average value of the dependent variable (see lecture 2) Y ˆ = Y
18 Remember that OLS predicts the mean or average value of the dependent variable (see lecture 2) Y ˆ = Y So in the case of a regression model with log wages as the dependent variable, LnW = b0 + b1age + b2male
19 Remember that OLS predicts the mean or average value of the dependent variable (see lecture 2) Y ˆ = Y So in the case of a regression model with log wages as the dependent variable, LnW = b0 + b1age + b2male the average of the fitted values equals the average of log wages _ ^ _ Ln ( W) = LnW
20 So the (average) difference in pay between men and women is then LnW men LnW women
21 So the (average) difference in pay between men and women is then LnW men LnW women = _ ^ LnW men LnW _ ^ women
22 So the (average) difference in pay between men and women is then LnW men LnW women = _ ^ LnW men LnW _ ^ women ^ ^ ^ ^ ^ = b0 + b1 Age + b2 b0 + b1 Age
23 The (average) difference in pay between men and women is then LnW men LnW women = _ ^ LnW men LnW _ ^ women = = ^ ^ ^ ^ ^ b0 + b1 Age + b2 b0 + b1 ^ b2 Age which is just the coefficient on the male dummy variable
24 The (average) difference in pay between men and women is then _ ^ LnW men _ ^ women LnW men LnW women = LnW ^ ^ ^ ^ ^ = b0 + b1 Age + b2 b0 + b1 Age ^ = b2 which is just the coefficient on the male dummy variable It also follows that the constant, b0, measures the intercept of default group (women) with age set to zero and b0 + b2 is the intercept for men
25 The (average) difference in pay between men and women is then _ ^ LnW men _ ^ women LnW men LnW women = LnW ^ ^ ^ ^ ^ ^ = b 0 + b1 Age + b2 b0 + b1 Age + b2 ^ = b2 which is just the coefficient on the male dummy variable
26 So the coefficients on dummy variables measure the average difference between the group coded with the value 1 and the group coded with the value 0 (the default or base group )
27 It also follows that the constant, b0, now measures the notional value of the dependent variable (in this case log wages) of the default group (in this case women) with age set to zero and b0 + b2 is the intercept and notional value of log wages at age zero for men
28 So to measure average difference between two groups LnW = β 0 + β 1 Group Dummy LnW Men B 0 +B 1 Β 1 Women B 0
29 A simple regression of the log of hourly wages on age using the data set ps4data.dta gives. reg lhwage age Source SS df MS Number of obs = F( 1, 12096) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhwage Coef. Std. Err. t P> t [95% Conf. Interval] age _cons Now introduce a male dummy variable (1= male, 0 otherwise) as an intercept dummy. This specification says the slope effect (of age) is the same for men and women, but that the intercept (or the average difference in pay between men and women) is different. reg lhw age male Source SS df MS Number of obs = F( 2, 12095) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhw Coef. Std. Err. t P> t [95% Conf. Interval] age male _cons Hence average wage difference between men and women =(b 0 (b 0 + b 2 ) = b 2 = 25% more on average
30 Note that if we define a dummy variables as female (1= female, 0 otherwise) then. reg lhwage age female Source SS df MS Number of obs = F( 2, 12095) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhwage Coef. Std. Err. t P> t [95% Conf. Interval] age female _cons The coefficient estimate on the dummy variable is the same but the sign of the effect is reversed (now negative). This is because the reference (default) category in this regression is now men Model is now LnW = b 0 + b 1 Age + b 2 female so constant, b 0, measures average earnings of default group (men) and b 0 + b 2 is average earnings of women So now average wage difference between men and women =(b 0 (b 0 + b 2 ) = b 2 = -25% less on average Hence it does not matter which way the dummy variable is defined as long as you are clear as to the appropriate reference category.
31 2) To measure Difference in Slope Effects between two groups LnW = β 0 + β 1 Group Dummy*Slope Variable LnW Men β 1 *Age 2 β 1 *Age 1 Women Age 1 Age 2 Age (Dummy Variable Interaction Term)
32 We need to consider an interaction term multiply slope variable (age) by dummy variable.
33 Now consider an interaction term multiply slope variable (age) by dummy variable. Model is now LnW = b 0 + b 1 Age + b 2 Female*Age
34 Now consider an interaction term multiply slope variable (age) by dummy variable. Model is now LnW = b 0 + b 1 Age + b 2 Female*Age This means that (slope) age effect is different for the 2 groups
35 Now consider an interaction term multiply slope variable (age) by dummy variable. Model is now LnW = b 0 + b 1 Age + b 2 Female*Age This means that (slope) age effect is different for the 2 groups dlnw/dage = b 1 if female = 0
36 Now consider an interaction term multiply slope variable (age) by dummy variable. Model is now LnW = b 0 + b 1 Age + b 2 Female*Age This means that (slope) age effect is different for the 2 groups dlnw/dage = b 1 if female = 0 = b 1 + b 2 if female = 1
37 . g femage=female*age /* command to create interaction term */. reg lhwage age femage Source SS df MS Number of obs = F( 2, 12095) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhwage Coef. Std. Err. t P> t [95% Conf. Interval] age femage _cons So effect of 1 extra year of age on earnings =.0097 if male = ( ) if female Can include both an intercept and a slope dummy variable in the same regression to decide whether differences were caused by differences in intercepts or the slope variables. reg lhwage age female femage Source SS df MS Number of obs = F( 3, 12094) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhwage Coef. Std. Err. t P> t [95% Conf. Interval] age female femage _cons In this example the average differences in pay between men and women appear to be driven by factors which cause the slopes to differ (ie the rewards to extra years of experience are much lower for women than men)- Note that this model is equivalent to running separate regressions for men and women since allowing both intercept and slope to vary
38 Using & Understanding Dummy Variables To measure average difference between two groups LnW = β 0 + β 1 Group Dummy LnW Men Β 1 Women
39 2) To measure Difference in Slope Effects between two groups LnW = β 0 + β 1 Group Dummy*Slope Variable LnW Men β 1 *Age 2 β 1 *Age 1 Women Age 1 Age 2 Age (Dummy Variable Interaction Term)
40 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories
41 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category
42 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South)
43 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South) Then define D North = 1 if live in the North, 0 otherwise
44 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South) Then define D North D Midlands = 1 if live in the North, 0 otherwise = 1 if live in the Midlands, 0 otherwise
45 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South) Then define D North D Midlands D South = 1 if live in the North, 0 otherwise = 1 if live in the Midlands, 0 otherwise = 1 if live in the South, 0 otherwise
46 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South) Then define D North D Midlands D South = 1 if live in the North, 0 otherwise = 1 if live in the Midlands, 0 otherwise = 1 if live in the South, 0 otherwise However
47 The Dummy Variable Trap The general principle of dummy variables can be extended to cases where there are several (but not infinite) discrete groups/categories In general just define a dummy variable for each category Eg if no. groups = 3 (North, Midlands, South) Then define D North D Midlands D South = 1 if live in the North, 0 otherwise = 1 if live in the Midlands, 0 otherwise = 1 if live in the South, 0 otherwise However As a rule should always include one less dummy variable in the model than there are categories, otherwise will introduce multicolinearity into the model
48 Example of Dummy Variable Trap Suppose interested in estimating the effect of (5) different qualifications on pay A regression of the log of hourly earnings on dummy variables for each of 5 education categories gives the following output. reg lhwage age postgrad grad highint low none Source SS df MS Number of obs = F( 5, 12092) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhwage Coef. Std. Err. t P> t [95% Conf. Interval] age postgrad (dropped) grad highint low none _cons
49 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications)
50 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the data set.
51 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the data set. Obs. Constant postgrad grad higher low noquals Sum
52 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum
53 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum
54 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum
55 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum
56 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum
57 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum Given the presence of a constant using 5 dummy variables leads to pure multicolinearity, becuse the sum=1 which is the same as the value of the constant)
58 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the regression data set. Obs. Constant postgrad grad higher low noquals Sum Given the presence of a constant using 5 dummy variables leads to pure multicolinearity, (the sum=1 = value of the constant)
59 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the data set. Obs. Constant postgrad grad higher low noquals Sum Given the presence of a constant using 5 dummy variables leads to pure multicolinearity, (the sum=1 = value of the constant)
60 Since in this example there are 5 possible education categories (postgrad, graduate, higher intermediate, low and no qualifications) 5 dummy variables exhaust the set of possible categories, so the sum of these 5 dummy variables is always one for each observation in the data set. Obs. constant postgrad grad higher low noquals Sum Given the presence of a constant using 5 dummy variables leads to pure multicolinearity, (the sum=1 = value of the constant) So can t include all 5 dummies and the constant in the same model
61 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set.
62 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low
63 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies
64 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies
65 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies
66 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies
67 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies and so the sum is no longer collinear with the constant
68 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies and so the sum is no longer collinear with the constant Doesn t matter which one you drop, though convention says drop the dummy variable corresponding to the most common category.
69 Solution: drop one of the dummy variables. Then sum will no longer equal one for every observation in the data set. Suppose drop the no quals dummy in the example above, the model is then Obs. Constant postgrad grad higher low Sum of dummies and so the sum is no longer collinear with the constant Doesn t matter which one you drop, though convention says drop the dummy variable corresponding to the most common category. However changing the default category does change the coefficients, since all dummy variables are measured relative to this default reference category
70 Example: Dropping the postgraduate dummy (which Stata did automatically before when faced with the dummy variable trap) just replicates the above results. All the education dummy variables pay effects are measured relative to the missing postgraduate dummy variable (which effectively is now picked up by the constant term). reg lhw age grad highint low none Source SS df MS Number of obs = F( 5, 12092) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhw Coef. Std. Err. t P> t [95% Conf. Interval] age grad highint low none _cons coefficients on education dummies are all negative since all categories earn less than the default group of postgraduates Changing the default category to the no qualifications group gives. reg lhw age postgrad grad highint low Source SS df MS Number of obs = F( 5, 12092) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lhw Coef. Std. Err. t P> t [95% Conf. Interval] age postgrad grad highint low _cons and now the coefficients are all positive (relative to those with no quals.)
71 Dummy Variables and Policy Analysis One important practical use of a regression is to try and evaluate the treatment effect of a policy intervention.
72 3) To Measure Effects of Change in the Average Behaviour of two groups, one subject to a policy the other not (the Difference-in-Difference Estimator) LnW Men Women Β 3 Treatment/Policy affects only a sub-section of the population Eg A drug, EMA, Change in Tuition Fees, Minimum Wage and may lead to a change in behaviour for the treated group - as captured by a change in the intercept (or slope) after the intervention (treatment) takes place
73 Dummy Variables and Policy Analysis One important practical use of a regression is to try and evaluate the treatment effect of a policy intervention. Usually this means comparing outcomes for those affected by a policy that is of concern to economists Eg a law on taxing cars in central London creates a treatment group, (eg those who drive in London) and those not, (the control group). Other examples targeted tax cuts, minimum wages, area variation in schooling practices, policing
74 In principle one could set up a dummy variable to denote membership of the treatment group (or not) and run the following regression LnW = a + b*treatment Dummy + u (1)
75 In principle one could set up a dummy variable to denote membership of the treatment group (or not) and run the following regression LnW = a + b*treatment Dummy + u (1) where Treatment = 1 if exposed to a treatment = 0 if not reg price newham if time>3 & (newham==1 croydon==1) reg price newham if time<=3 & (newham==1 croydon==1) reg price newham after afternew if time>3 & (newham==1 croydon==1)
76 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect.
77 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect. This is because there may always have been a different value for the treatment group even before the policy intervention took place
78 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect. This is because there may always have been a different value for the treatment group even before the policy intervention took place ie. Could estimate LnW = a + b*treatment Dummy + u in the period before any treatment took place
79 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect. This is because there may always have been a different value for the treatment group even before the policy intervention took place ie. Could estimate LnW = a + b*treatment Dummy + u in the period before any treatment took place but what ever the effect observed it cannot be caused by the treatment since these events are observed before the treatment took place.
80 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect. This is because there may always have been a different value for the treatment group even before the policy intervention took place ie. Could estimate LnW = a + b*treatment Dummy + u in the period before any treatment took place but what ever the effect observed it cannot be caused by the treatment since these events are observed before the treatment took place. The idea instead is then to compare the change in Y for the treatment group who experienced the shock with the change in Y of the control group who did not
81 Problem: a single period regression of the dependent variable on the treatment variable as in (1) will not give the desired treatment effect. This is because there may always have been a different value for the treatment group even before the policy intervention took place ie. Could estimate LnW = a + b*treatment Dummy + u in the period before any treatment took place but what ever the effect observed it cannot be caused by the treatment since these events are observed before the treatment took place. The idea instead is then to compare the change in Y for the treatment group who experienced the shock with the change in Y of the control group who did not - the difference in difference estimator
82 LnW Men Women Β 3
83 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences
84 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by
85 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences
86 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences
87 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences [Yt 2 Yt 1 ] - [Yc 2 Yc 1 ] = Effect of Policy + other influences
88 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences [Yt 2 Yt 1 ] - [Yc 2 Yc 1 ] = Effect of Policy + other influences - Effect of other influences
89 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences [Yt 2 Yt 1 ] - [Yc 2 Yc 1 ] = Effect of Policy + other influences - Effect of other influences = Effect of Policy
90 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences [Yt 2 Yt 1 ] - [Yc 2 Yc 1 ] = Effect of Policy + other influences - Effect of other influences = Effect of Policy (assuming the effect of other influences is the same for both treatment and control groups )
91 If the change for Treatment group reflects both [Yt 2 Yt 1 ] = Effect of Policy + other influences but the change for control group is only caused by [Yc 2 Yc 1 ] = Effect of other influences then the difference in differences [Yt 2 Yt 1 ] - [Yc 2 Yc 1 ] = Effect of Policy + other influences - Effect of other influences = Effect of Policy (assuming the effect of other influences is the same for both treatment and control groups ) Hence the need to try and choose a control group that is similar to the treatment group (apart from the experience of the treatment)
92 In practice this estimator can be obtained by combining (pooling) the data over the periods before and after and running the following regression LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy
93 In practice this estimator can be obtained by combining (pooling) the data over the periods before and after and running the following regression LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy where now After is a dummy variable = 1 if data observed after the treatment = 0 if data observed before the treatment
94 In practice this estimator can be obtained by combining (pooling) the data over the periods before and after and running the following regression LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy where now After is a dummy variable = 1 if data observed after the treatment = 0 if data observed before the treatment a is the average wage of the control group in the base year, a2, is the average wage of the control group in the second year, b1 gives the difference on wages between treatment and control group in the base year b2 is the difference in difference estimator the change in wages for the treatment group relative to the control in the second period. Why? LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy
95 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0
96 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0
97 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a
98 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1
99 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2
100 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2
101 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is
102 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2)
103 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1)
104 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2
105 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a )
106 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a ) = a2 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0
107 Similarly = a If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a ) = a2 so the difference in difference estimator = Change in wages for treatment change in wages for control = (a2 +b2) - ( a2 ) = b2
108 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a ) = a2 so the difference in difference estimator = Change in wages for treatment change in wages for control
109 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a ) = a2 so the difference in difference estimator = Change in wages for treatment change in wages for control = (a2 +b2) - ( a2 )
110 LnW = a + a2after + b1treatment Dummy + b2after*treatment Dummy If After=0 and Treatment Dummy = 0 LnW = a + a2*0 +b1*0 + b2*0 = a Similarly If After =0 and Treatment Dummy = 1, LnW = a + b1 If After =1 and Treatment Dummy = 0, LnW = a + a2 If After =0 and Treatment Dummy = 1, LnW = a + a2 +b1 + b2 So the change in wages for the treatment group is (a + a2 +b1 + b2) (a + b1) = a2 +b2 and the change in wages for the control group is (a + a2 ) (a ) = a2 so the difference in difference estimator = Change in wages for treatment change in wages for control = (a2 +b2) - ( a2 ) = b2
111
112 Example In July 2005, London won the rights to host the 2012 Olympic games. Shortly afterward there were media reports that house prices were rising fast in areas close to the Olympic site. Can evaluate whether this was true by using Newham as the Treatment area (the borough in which the Olympic site is located) and a similar London borough further away from the site as a control. The data set olympics.dta has monthly data on house prices over time in Newham & Hounslow. The dummy variable newham takes the value 1 if the house price observation is from Newham and 0 if not. The dummy variable after takes the value 1 if the month was after the Olympic announcement and 0 otherwise. The interaction term afternew g afternew=after*newham takes the value 1 only if the month is after the event and the observation is in Newham. The coefficient on this term will be the difference-in-difference estimator (the differential effect of the Olympic bid on house prices in newham relative to the control area of croydon.. reg price after if newham==1 Source SS df MS Number of obs = F( 1, 79) = Model e e+10 Prob > F = Residual e e+09 R-squared = Adj R-squared = Total e e+09 Root MSE = price Coef. Std. Err. t P> t [95% Conf. Interval] after _cons House prices were indeed higher in Newham after the Olympic announcement, but. reg price after if hounslow==1 Source SS df MS Number of obs = F( 1, 78) = Model e e+10 Prob > F = Residual e R-squared = Adj R-squared = Total e e+09 Root MSE = price Coef. Std. Err. t P> t [95% Conf. Interval]
113 after _cons they were also higher in Hounslow Moreover the annual rate of growth of house prices (approximated by the log of the 12 month change) was not significantly different in the after period. reg dlogp after if newham==1 Source SS df MS Number of obs = F( 1, 70) = 0.74 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = dlogp Coef. Std. Err. t P> t [95% Conf. Interval] after _cons and the difference-in-difference analysis confirms that there was no differential house price growth between the two areas. It seems claims of a house price effect were exaggerated. reg logp after newham afternew if newham==1 hounslow==1 Source SS df MS Number of obs = F( 3, 157) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = logp Coef. Std. Err. t P> t [95% Conf. Interval] after newham afternew _cons
114 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data
115
116 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data The idea is to include a set of dummy variables for each quarter (or month or day) which will then net out the average change in a variable resulting from any seasonal fluctuations
117 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data The idea is to include a set of dummy variables for each quarter (or month or day) which will then net out the average change in a variable resulting from any seasonal fluctuations Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 X + u t
118 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data The idea is to include a set of dummy variables for each quarter (or month or day) which will then net out the average change in a variable resulting from any seasonal fluctuations Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 X + u t Hence the coefficient on the quarterly dummy Q1 =1 if data belong to the 1 st quarter of the year (Jan-Mar) = 0 otherwise
119 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data The idea is to include a set of dummy variables for each quarter (or month or day) which will then net out the average change in a variable resulting from any seasonal fluctuations Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 X + u t Hence the coefficient on the quarterly dummy Q1 =1 if data belong to the 1 st quarter of the year (Jan-Mar) = 0 otherwise gives the level of Y in the 1 st quarter of the year relative to the constant (Q4 level of Y) averaged over all Q1 observations in the data set
120 Using Dummy Variables to capture Seasonality in Data Can also use dummy variables to pick out and control for seasonal variation in data The idea is to include a set of dummy variables for each quarter (or month or day) which will then net out the average change in a variable resulting from any seasonal fluctuations Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 X + u t Hence the coefficient on the quarterly dummy Q1 =1 if data belong to the 1 st quarter of the year (Jan-Mar) = 0 otherwise gives the level of Y in the 1 st quarter of the year relative to the constant (Q4 level of Y) averaged over all Q1 observations in the data set Series net of seasonal effects are said to be seasonally adjusted
121 It may also be useful to model an economic series as a combination of seasonal and a trend component
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123 It may also be useful to model an economic series as a combination of seasonal and a trend component Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 Trend + u t where Trend = 1 in year 1
124 It may also be useful to model an economic series as a combination of seasonal and a trend component Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 Trend + u t where Trend = 1 in year 1 = 2 in year 2
125 It may also be useful to model an economic series as a combination of seasonal and a trend component Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 Trend + u t where Trend =1 in year 1 = 2 in year 2 = T in year T
126 It may also be useful to model an economic series as a combination of seasonal and a trend component Y t = b 0 + b 1 Q1 + b 2 Q2+ b 3 Q3 + b 4 Trend + u t where Trend =1 in year 1 = 2 in year 2 = T in year T since dy t /dtrend = b 4 given that the coefficient measures the unit change in y for a unit change in the trend variable and the units of measurement in this case are years
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