One time series tends toward the (linear) function of another time series, how to find that function?

I have two time series

• $p_t$, the daily market price of a particular kind of good
• $f_t$, the daily production of such good

Now assume that there is a unique relationship that tells you the optimal production given a price (a supply function):

$q(p_t) = a + b p_t$.

The production series $f_t$ approaches $q(p_t)$ but does so slowly and noisily. An example of what I mean by that is:

I have to estimate $a$ and $b$ by observing only $f_t$ and $p_t$. I am sure it can be done, but I am really rusty in time series. Thanks for the help!

• Isn't the production function equal to the price function with a smaller intercept and a white noise error term? Like $$q(f_t)=c+bp_t+\epsilon_t$$, where $\epsilon_t$ is a white noise error term and $c$ is another intercept. – fredrikhs Aug 6 '13 at 21:20
• In this run, this is superficially true. But I know (since this data is from simulation) that there is a supply function that $f_t$ tends to with some delay.Thank you! – CarrKnight Aug 7 '13 at 6:26
• Looks like something like $$q(f_t)=c+b_1p_{t-1}+b_2p_{t-2}+..+b_mp_{t-m}+t^{-1}+\epsilon_t$$, where $m$ is the lag length. – fredrikhs Aug 7 '13 at 7:00

You could test your bivariate time series for a presence of cointegration.

Here a simplified definition: two individually integrated (non-stationary) time series are cointegrated, if there exists a linear combination of these, which is integrated of order zero (i.e. stationary). Cointegrated variables are characterised through a long run equilibrium. There are random, stationary deviations from this equlibrium, but the variables return to their equilibrium in the long run. So they share a common stochastic trend.

In order to test for the existence of cointegration, you first regress one variable on the other, e.g.

$$f_t=\alpha*p_t+z_t$$ Where $f_t$ and $p_t$ have the same definition as yours. $\alpha$ is a OLS estimator of the slope coefficient and $z_t$ represents error term.

You then take the estimators of the error term, $\hat{z_t}$, and check if they are stationary, e.g. using Dickey-Fuller-Test, Durbin-Watson-Test, etc. If you can reject the null hypothesis of nonstationarity of the error terms, you can conclude that there exists a cointegration relation betwenn your two variables. The cointegrating vector is then: (1, $\ -\alpha$)

Write $f_t^*$ for the optimal production level. You already assume that this is a deterministic function of price, $f_t^* = a+bp_t$. Next, you have to assume a relation between optimal and actual production - and what will be the model to estimate $a$ and $b$ will depend on this assumption. You say that you now that actual production tends to optimal production with some delay and noisily. As user @fredrikhs essentially pointed out, one way to describe this is to specify $f_t= g\left(f_t^*, \frac ct\right) + \epsilon$ say simply , $f_t= f_t^* +\frac ct +\epsilon_t, \epsilon_t=$ white noise. Substitute the deterministic relation for $f_t^*$ into this to get $$f_t =a+bp_t+\frac ct +\epsilon_t$$ You have observations on $f_t$ and on $p_t$ and you know $t$- you can run a regression to estimate $a$ , $b$ and $c$.

Then you can start worrying about the quality and reliability of your results, these being time series, obviously non-stationary for the first 3.000 days , etc.