How to explain negative correlation between quantity sold and expenditure on advertisement? I have received the following dataset from our economics Professor. It has 15 observations and 4 variables - 'qsold' (quantity sold of product X), psn (price of X), pcb (price of a substitute product Y), adv (expenditure on advertising of X). I am supposed to derive a demand function (qsold = B0 + B1 (psn) + B2 (pcb) + B3 (adv)). Now theoretically, all three independent variables are supposed to have a relationship with qsold, however, I am supposed to explore only linear relationship, so, I tried to fit the following model.
df1
qsold   psn pcb adv
1183    1361.97 1405.78 3.22
974 1520.49 1369.17 3.39
1179    1361.43 1448.71 4.03
1258    1159.67 1465.12 3.91
1161    1297.74 1383.93 3.46
1052    1362.44 1450    3.64
992 1447.25 1404.4  3.55
1213    1316.93 1418.03 3.81
1133    1365.97 1391.95 4.21
1001    1283.92 1403.11 4.22
1221    1329.34 1428.9  3.38
1137    1278.41 1426.81 3.89
1112    1466.21 1442.68 3.65
1025    1355.73 1359.79 4.25
1277    1377.06 1455.03 3.35

lm(formula = qsold ~ psn + pcb + adv, data = df1)

Residuals:
    Min      1Q  Median      3Q     Max 
-118.47  -31.59   12.42   39.46   92.43 

Coefficients:
             Estimate Std. Error t value Pr(&gt;|t|)  
(Intercept)  635.4451  1240.7873   0.512   0.6187  
psn           -0.5897     0.2647  -2.228   0.0477 *
pcb            1.1835     0.6650   1.780   0.1027  
adv         -103.7231    62.7722  -1.652   0.1267  
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 72.57 on 11 degrees of freedom
Multiple R-squared:  0.5764,    Adjusted R-squared:  0.4609 
F-statistic:  4.99 on 3 and 11 DF,  p-value: 0.02004


In the output above, only psn is significant (based on t stats). Our Professor told us that we should consider only significant coefficients in the demand function. I am finding it difficult to agree with. In this case, if I consider only psn in my demand function, I am perhaps violating the null hypothesis, which is based on F stats (that all coefficients are 0). Also, if I consider only psn I am essentially negating the combined effect of all three variables, and basically choosing a different model that what I fitted.

adv has a negative coefficient, which is against my theoretical understanding. As I have read that it has a positive correlation with the quantity demanded. How can I explain this relationship?


If possible please cite literature, which I can refer. Also, please note collecting data for other variables, increasing sample size, or fitting non-linear models is not a possibility.
In the output above, only psn is significant (based on t stats). Our Professor told us that we should consider only significant coefficients in the demand function. I am finding it difficult to agree with. In this case, if I consider only psn in my demand function, I am perhaps violating the null hypothesis, which is based on F stats (that all coefficients are 0). Also, if I consider only psn I am essentially negating the combined effect of all three variables, and basically choosing a different model that what I fitted.

It is important to take into account significance because you cannot directly observe data generating process. Regression does not tell you what $\beta_0, \beta_1 ... $ are, it tells you what they appear to be based on fitting line that minimizes the sum of squared residuals through data. Yes in expectations (provided all OLS assumptions hold) $E[\hat{\beta}] = \beta$ but this result only holds in the limit as the number of observations go to infinity (see discussion in Verbeek. A Guide to Modern Econometrics or really any econometrics textbooks for more details).
In this particular case the $F$-stat (with 3 an 11 degrees of freedom) is 4.99 which is sufficient to reject the null of joint insignificance i.e. $H_0: \beta_0 = \beta_1 = ... \beta_k = 0$ so that is not a problem.
However, you cannot reject the null hypothesis that $\beta_3$ (effect of advertising) is zero. Consequently, just taking the results of the regression you present at face value (which is not necessary always appropriate in real research but might be in classroom setting), the correct answer would be &quot;advertising has no significant effect on quantity sold, we cannot reject the hypothesis that the true effect of advertising is 0&quot;.

adv has a negative coefficient, which is against my theoretical understanding. As I have read that it has a positive correlation with the quantity demanded. How can I explain this relationship?

This could be because you have some reverse causality. Demand equations are endogenous systems and should be modeled with simultaneous equations, not just simple regression (see this explanation for how to do that in R). Using simple multivariate regression in such cases is likely going to result in biased coefficients.
Possibly when demand is low firms ramp up spending on advertising, while when demand is high and firms quickly sell their inventory they might  not want to spend extra money on advertising. This creates reverse causality issue which biases the $\beta_3$ estimates that no longer give you accurate effect of adv  on $y$ (quantity sold). It is possible that advertising spending increases sales, but higher sales are simply associated with less spending and the second effect from sales $\rightarrow$ adv simply dominates.
This being said you mention that this is an exercise imposed on you by your teacher not real research. It is possible your teacher just wants to test if you can interpret coefficient estimates properly not if you understand concepts such as reverse causality (if you did not covered this in your class then probably your teacher is not interested in discussion of that). You should ask your teacher about their expectations regarding the answer.