## AR(1) model: Tesla stock versus Tesla return

**Question**. You run two AR(1) regressions: 1) for Tesla stock price , , and 2) for its return , . Here the errors are i.i.d. normal with mean and variance Based on the 5-year chart of the stock price below (see Chart 1), what would be your expectations about the coefficients and

**Answer**. Suppose that instead of the time series model we have a simple regression and on the stock chart we have the values of on the horizontal axis and the values of 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 , we can see that both and must be positive (see Chart 2). The same intuition applies to the time series model

Table 1 contains estimation results for the first model.

Coefficient | Estimate | p-value |

152.282 | 0.023 | |

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.

The straight line that approximates the cloud of observed pairs should be very close to the axis. That is, both and should be very close to zero.

See estimation results in Table 2.

Coefficient | Estimate | p-value |

0.0018 | 0.106 | |

-0.0056 | 0.795 |

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