4
Mar 24

AR(1) model: Tesla stock versus Tesla return

AR(1) model: Tesla stock versus Tesla return

Question. You run two AR(1) regressions: 1) for Tesla stock price Y_{t}, Y_{t}=\alpha +\beta Y_{t-1}+\varepsilon _{t}, and 2) for its return R_{t}R_{t}=\phi +\psi R_{t-1}+\delta _{t}. Here the errors \varepsilon _{t},\delta _{t} are i.i.d. normal with mean 0 and variance \sigma ^{2}. Based on the 5-year chart of the stock price below (see Chart 1), what would be your expectations about the coefficients \alpha , \beta , \phi , and \psi ?

Chart 1. 5-year chart of TSLA stock. Source: barchart.com

Answer. Suppose that instead of the time series model Y_{t}=\alpha +\beta Y_{t-1}+\varepsilon _{t} we have a simple regression Y_{t}=\alpha +\beta X_{t}+\varepsilon _{t} and on the stock chart we have the values of X_{t} on the horizontal axis and the values of Y_{t} on the vertical axis. Then instead of the time series chart we would have a scatterplot. Drawing a straight line to approximate the cloud of observed pairs \left(X_{t},Y_{t}\right) , we can see that both \alpha and \beta must be positive (see Chart 2). The same intuition applies to the time series model Y_{t}=\alpha +\beta Y_{t-1}+\varepsilon _{t}.

Chart 2. Same chart of Tesla stock viewed as a scatterplot with fitted line

Table 1 contains estimation results for the first model.

Table 1
Coefficient Estimate p-value
\alpha 152.282 0.023
\beta 0.9973 0.000

 

The fundamental difference between the stock and its return is that the return cannot be trending for extended periods of time. The intuition is that if, for example, the return for some stock is persistently positive, then everybody starts investing in it and seeing sizable profits. However, the paper profits must be realized sooner or later, which means investors at some point will start selling the stock and the return becomes negative. As a result, the return must oscillate around zero. This intuition is confirmed in Chart 3, which displays the return for Tesla stock, and in Chart 4, which is a nonparametric estimation of the density of that return.

Chart 3. Chart for return on Tesla, from Stata

The straight line that approximates the cloud of observed pairs \left(R_{t-1},R_{t}\right) should be very close to the x axis. That is, both \phi and \psi should be very close to zero.

Chart 4. The density of return on Tesla is centered almost at zero

See estimation results in Table 2.

Table 2
Coefficient Estimate p-value
\phi 0.0018 0.106
\psi -0.0056 0.795