# Demand estimation with a lagged dependent variable

Suppose I have the following structural equation for demand estimation in time-series:

$$q_t=\beta_0+\beta_1\hat{p_t}+\beta_2incom_t+\beta_3q_{t-1}+\epsilon_t$$

Where $q_t$ stands for the quantity of the product, $\hat{p}$ stands for the instrumented price with supply shifters, $incom_t$ stands for income, $q_{t-1}$ stands for the lagged quantity and $\epsilon_t$ are the random residuals. All the betas are estimated coefficients -- say, through two-stage least squares. All variables are log-transformed too.

Suppose also that the lagged variable seems to solve the problem of autocorrelation between the residuals and its coefficient is also significant. Suppose also that the model passes through all the necessary tests for a viable two-stage least squares estimation -- the sargan test is ok, the instruments are strong in the first stage etc.

Knowing this, what are the possible problems of having a lagged variable in your estimation? Does it change any interpretation of the elasticity ($\beta_1$)? I understand that I could also have some sort of long-run inferred elasticity if I set $q_t=q_{t-1}$ after estimating the coefficients of the equation (Am I wrong?).

Does it change any interpretation of the elasticity ($β_1$)?

You walk into the firm where you work as an analyst, and the Sales Director calls and asks "I want to raise the price $10\%$ today. How will demand be affected in percentage terms?

Well, you don't expect income to have changed from one day to the next, and what was demanded yesterday -it was yesterday. So the only variable (for today) in the equation is price, and the best you can say is "I expect a $(10 \times \hat \beta_1) \%$ effect".

So it is the short term price elasticity of demand, irrespective of the fact that the level of demand is determined by other factors also, lagged demand included.

I understand that I could also have some sort of long-run inferred elasticity if I set $q_t=q_{t−1}$ after estimating the coefficients of the equation.

Yes, either in an old fashioned "deterministic", $q_t=q_{t−1}$ way, or in a stochastic, $E(q_t) = E(q_{t−1})$ "mean-stationary" way.

Assume now that demand is already mean-stationary according to your analysis , and you get the same question from the Sales Director as before. Will your answer change compared to the previous one?

The answer to the Sales Director should not change, because you can condition on the given and known past, for a more accurate (= focused on the specific situation) answer so you use $\hat E(q_t \mid p_t, I_t, q_{t-1}) = \hat \beta_0+ \hat \beta_1\hat{p_t}+\hat \beta_2I_t+\hat \beta_3q_{t-1}$ and not the unconditional (estimated) relation $\hat E(q_t) = \hat \beta_0+ \hat \beta_1E[\hat{p_t}]+\hat \beta_2E[I_t]+\hat \beta_3E[q_{t-1}]$,which would lead to the long-run (unconditional) elasticity. But if he was to ask, say, "I want to introduce the product to a new market where the consumers are comparable in income to what we already have. What demand-responsiveness to price should I have in mind?", then the best you could do is provide the long-run elasticity.
• Do you mean $10 * \hat{\beta_1}$ effect as opposed to $10*\hat{\beta_2}$? – Hessian Jun 5 '15 at 15:56