1. assignment 8 solutions ► yogurt brands Developed for
Managerial Economics &
Decision Sciences Department
Developed for
assignment 8
solutions
► yogurt brands
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2. fixed effects ► STATA ► cross section and panel data ► fixed effects
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
learning
objectives
► STATA
 fixed effects regression: xi:regress
► cross section and panel data
 working with data across years
 regression for panel data
► fixed effects
 definition
 use of fixed effects to eliminate ovb
readings
► (KTN) Fixed Effects
► (CS) YogurtFE
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3. ln(sales1) = b0 + b1·ln(price1) + b2·promo1
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
yogurt brands ◄
yogurt brands
i. elasticity (brand one). We know that the log-log specification yields directly the estimate of the elasticity, thus we run the regression
ln(sales1) = b0 + b1·ln(price1) + b2·promo1
. regress lsales1 lprice1 promo1
lsales1 | Coef. Std. Err t P>|t| [95% Conf. Interval]
lprice1 |
promo1 |
_cons |
► We find elasticity to be –1.22 which means that an increase with one percent in price leads to a decrease in sales with 1.22 percent. The sign is correct but we do not have any indication of whether the magnitude is correct or not. Of course, along with a myriad of problems such as heteroskedasticity, the omitted variable bias and a miss-specified curvature are the most “dangerous” ones.
► Here we focus on omitted variables bias since we are given a panel data and we can apply the fixed effects model: we have to find the characteristic for which across time the omitted variable has a constant (fixed) effect. The candidate is store: we assume that whatever variable is omitted it would affect in the same way brand one within each store but differently across stores…
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page | 1

4. fixed effects yogurt brands
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
yogurt brands ◄
yogurt brands
ii & iii. elasticity (brand one). We know that the log-log specification yields directly the estimate of the elasticity, thus we run the regression with fixed effects (by store):
ln(sales1) = a0 + a1·dstore11 + a2·dstore14 + a3·dstore15 + b1·ln(price1) + b2·promo1
. xi:regress lsales1 lprice1 promo1 i.store
i.store _Istore_ (naturally coded; _Istore_5 omitted)
lsales1 | Coef. Std. Err t P>|t| [95% Conf. Interval]
lprice1 |
promo1 |
_Istore_11 |
_Istore_14 |
_Istore_15 |
_cons |
► We find elasticity to be –1.79 thus below the previous estimate. Since this figure is found as a result of fixed effects it might be an indication that the previous regression leads to an overestimation of the elasticity.
► What can be the omitted variable that can explain this behavior? In this case the story telling has to be consistent to the assumption that the omitted variable has the same effect across time within each store but to be different between stores… A few come to mind: location, management…
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page | 2

5. ln(sales3) = b0 + b1·ln(price3) + b2·promo3
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
yogurt brands ◄
yogurt brands
iv. elasticity (brand three). The initial regression (no fixed effects included):
ln(sales3) = b0 + b1·ln(price3) + b2·promo3
. regress lsales3 lprice3 promo3
lsales3 | Coef. Std. Err t P>|t| [95% Conf. Interval]
lprice3 |
promo3 |
_cons |
► We find elasticity to be –2.146 which means that an increase with one percent in price leads to a decrease in sales with percent. The sign is correct but we do not have any indication of whether the magnitude is correct or not. We look again for omitted variables bias by using the fixed effects model.
© Kellogg School of Management
page | 3

6. fixed effects yogurt brands
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
yogurt brands ◄
yogurt brands
iv. elasticity (brand three). The regression with fixed effects (by store):
ln(sales3) = a0 + a1·dstore11 + a2·dstore14 + a3·dstore15 + b1·ln(price3) + b2·promo3
. xi:regress lsales3 lprice3 promo3 i.store
i.store _Istore_ (naturally coded; _Istore_5 omitted)
lsales3 | Coef. Std. Err t P>|t| [95% Conf. Interval]
lprice3 |
promo3 |
_Istore_11 |
_Istore_14 |
_Istore_15 |
_cons |
► We find elasticity to be –2.073 thus below the previous estimate. Since this figure is found as a result of fixed effects it might be an indication that the previous regression leads to an underestimation of the elasticity.
© Kellogg School of Management
page | 4

7. fixed effects yogurt brands
Managerial Economics &
Decision Sciences Department
assignment 8 - solutions
fixed effects
yogurt brands ◄
yogurt brands
bonus: interpretation of coefficient on dummy variables. The regression with fixed effects (by store):
ln(sales3) = a0 + a1·dstore11 + a2·dstore14 + a3·dstore15 + b1·ln(price3) + b2·promo3
. xi:regress lsales3 lprice3 promo3 i.store
i.store _Istore_ (naturally coded; _Istore_5 omitted)
lsales3 | Coef. Std. Err t P>|t| [95% Conf. Interval]
lprice3 |
promo3 |
_Istore_11 |
_Istore_14 |
_Istore_15 |
_cons |
► _Istore_5 is the dummy variable omitted, thus the coefficients a1 = -0.27, a2 = 0.73 and a3 = should be interpreted relative to the omitted dummy:
a1 = represents the difference in sales when dstore11 = 1 vs. dstore5 = 1 (lower sales in store 11 compared to store 5)
a2 = 0.73 represents the difference in sales when dstore14 = 1 vs. dstore5 = 1 (higher sales in store 14 compared to store 5)
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page | 5