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I'm trying to fit an AR(p) process to the standardized, 10 years long time series of monthly logreturns of a stock index and get extremely poor fit. I'm not surprised, because if I had a good fit, then it would mean the return of the next period could be predicted as the linear combination of the returns from the previous p periods, which would be very strange and in addition it would contradict even to the weak form of the efficient market hypothesis.

I believe the following statements hold:

  1. Wold's decomposition theorem states any covariance-stationary time series $Y_t$ can be written as the sum of two time series, one deterministic ($\eta_t$) and one stochastic ($MA(\infty)=\sum_{j=0}^\infty b_j \varepsilon_{t-j}$): \begin{align*} Y_t=\sum_{j=0}^\infty b_j \varepsilon_{t-j}+\eta_t \end{align*} My log returns time series seem to be covariance-stationary by statistical tests.
  2. An $MA(\infty)$ model can be approximated by an $MA(q)$ model with arbitrary precision.
  3. An invertible $MA(q)$ model can be represented as an $AR(\infty)$ model.
  4. An $AR(\infty)$ model can be approximated by an $AR(p)$ model with arbitrary precision.

If all of the above are true, why can't I fit an $AR(p)$ process to the log returns time series? In other words, why doesn't the Wold's decomposition theorem imply a good AR(p) fit?

I have the following guesses:

  1. The deterministic term ($\eta_t$) is not zero in the Wold decomposition of the log returns time series. It doesn't seem probable to me.
  2. The $MA(\infty)$ model can be approximated by an $MA(q)$ model with arbitrary precision only under certain conditions which do not hold in my case.
  3. The $MA(q)$ model (that would approximate the $MA(\infty)$ model) is not invertible in my case.
  4. The $AR(\infty)$ model can be approximated by an $AR(p)$ model with arbitrary precision only under certain conditions which do not hold in my case.
  5. Maybe an $AR(p)$ model with good fit does exist, but the methods I used (ordinary least squares and maximum likelihood method) don't find it.

Could someone please point out the reason(s) from my guesses or from outside those why I don't get an $AR(p)$ model with good fit?

(In my VAR(p) model all of the common information criteria suggest using the order of p=1. Statistical tests show all of the 3 assumptions on the errors of my fitted VAR(1) process hold: $E(u_t)=0$ and $E(u_t^2|x_i)=const$ (i.e. no conditional heteroscedasticity), and $E(u_t, u_s)=0$ (i.e. no serial correlation). I know conditional heteroscedasticity is common in stock returns, that's why (G)ARCH model is preferred over AR(I)MA, but in my case there seems to be no conditional heteroscedasticity, and anyways, it shouldn't affect my question that why doesn't the Wold's decomposition theorem imply a good AR(p) fit.)

Thank you in advance for the answer.

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  • $\begingroup$ You haven’t shown any results. Why do you think it’s not a good fit? $\endgroup$ – The Laconic Apr 10 at 12:32
  • $\begingroup$ Because the coefficient of determination (R squared) is close to zero. $\endgroup$ – corres Apr 10 at 12:50
  • $\begingroup$ Isn’t that what you should expect? Didn’t you in fact fit an AR(p) model, but find that all the $b_j$ are indistinguishable from zero? $\endgroup$ – The Laconic Apr 10 at 13:17
  • $\begingroup$ Yes, that's the case. And that's in line with my expectations, why would I be able to fit AR(p) model and predict returns? Of course I can't do that. My problem is it looks like I don't understand or misunderstand the Wold decomposition theorem. In my understanding there should be an MA(inf) model that fits to the stock returns. Then an MA(q) model should exists that approximates the MA(inf). The MA(q) should be invertible to an AR(inf), and an AR(p) should exist that approximates the AR(inf). But my AR(p) fitting is poor. So where's the problem in my implication chain? $\endgroup$ – corres Apr 10 at 13:55
  • $\begingroup$ But you have an MA model that fits: $Y_t = \eta_t$, which is what you expected. $\endgroup$ – The Laconic Apr 10 at 19:30
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To summarize our exchange above:

  • Given the assumptions behind the Wold decomposition, and the efficient market hypothesis, you should expect to find that stock returns most closely follow a process with $b_0=1$, $b_i=0$ for $i>0$, and $\eta_t=0$.
  • This is just an MA(0) process.
  • We can express this trivially in AR($\infty$) form by just setting all the AR coefficients to zero.
  • We can approximate that arbitrarily closely as an AR(p) process with all the AR coefficients equal to zero.
  • This is a "good fit" in the sense that the model is consistent with the data, but not in the sense of having a high $R^2$. Which is fine, because the Wold decomposition doesn't have to yield a high $R^2$, nor should it.
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  • $\begingroup$ Very clear, this is an exhaustive answer to my question, thank you very much. $\endgroup$ – corres Apr 12 at 6:43

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