assignment 8 solutions ► yogurt brands Developed for Managerial Economics & Decision Sciences Department Developed for assignment 8 solutions ► yogurt brands © Kellogg School of Management

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 © Kellogg School of Management

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 | -1.221729 .2111436 -5.79 0.000 -1.637003 -.8064547 promo1 | .7136112 .1182778 6.03 0.000 .4809843 .9462381 _cons | 4.925993 .5297197 9.30 0.000 3.884149 5.967838 ► 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… © Kellogg School of Management page | 1

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_5-15 (naturally coded; _Istore_5 omitted) ------------------------------------------------------------------------------ lsales1 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lprice1 | -1.797195 .1167034 -15.40 0.000 -2.026732 -1.567658 promo1 | .5291696 .0647114 8.18 0.000 .4018925 .6564468 _Istore_11 | -.6406308 .0373443 -17.15 0.000 -.7140812 -.5671804 _Istore_14 | .4403146 .037269 11.81 0.000 .3670123 .5136169 _Istore_15 | -.0334855 .0371761 -0.90 0.368 -.1066052 .0396342 _cons | 3.544125 .2932429 12.09 0.000 2.967362 4.120888 ► 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… © Kellogg School of Management page | 2

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 | -2.146479 .2173114 -9.88 0.000 -2.573884 -1.719074 promo3 | .6372796 .2547515 2.50 0.013 .1362382 1.138321 _cons | .8359389 .6423563 1.30 0.194 -.4274374 2.099315 ► We find elasticity to be –2.146 which means that an increase with one percent in price leads to a decrease in sales with 2.146 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

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_5-15 (naturally coded; _Istore_5 omitted) ------------------------------------------------------------------------------ lsales3 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lprice3 | -2.073734 .1554244 -13.34 0.000 -2.37943 -1.768039 promo3 | .8190779 .1825961 4.49 0.000 .4599399 1.178216 _Istore_11 | -.2762774 .0583409 -4.74 0.000 -.3910249 -.1615299 _Istore_14 | .7389823 .0582163 12.69 0.000 .6244799 .8534847 _Istore_15 | -.0174693 .0582409 -0.30 0.764 -.13202 .0970814 _cons | .9368165 .461271 2.03 0.043 .0295684 1.844065 ► 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

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_5-15 (naturally coded; _Istore_5 omitted) ------------------------------------------------------------------------------ lsales3 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lprice3 | -2.073734 .1554244 -13.34 0.000 -2.37943 -1.768039 promo3 | .8190779 .1825961 4.49 0.000 .4599399 1.178216 _Istore_11 | -.2762774 .0583409 -4.74 0.000 -.3910249 -.1615299 _Istore_14 | .7389823 .0582163 12.69 0.000 .6244799 .8534847 _Istore_15 | -.0174693 .0582409 -0.30 0.764 -.13202 .0970814 _cons | .9368165 .461271 2.03 0.043 .0295684 1.844065 ► _Istore_5 is the dummy variable omitted, thus the coefficients a1 = -0.27, a2 = 0.73 and a3 = -0.01 should be interpreted relative to the omitted dummy: a1 = -0.27 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) © Kellogg School of Management page | 5