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I ran into an issue while trying to predict stock prices using a Vector Autoregression (VAR) model. After noticing that all the series are non-stationary (see example below): Plots of non-stationary series I took first differences of all variables, making them stationary (this result was confirmed by an Augmented Dickey Fuller test): Plots of stationary series However, when fitting a VAR(1) model, I noticed that forecasted values only capture the overall upward/downward trend of the stock price, without being able to predict whether it will go up or down. Forecast v Actual Values 1

On the other hand, if I arbitrarily crank up the number of lags and I fit a VAR(200) model, predictions are more accurate at least in predicting the movements of the stocks:

enter image description here

I have only provided one stock as reference, although the model contains 30 of them and their respective plots are fairly similar. Now, my question is: since I am quite sure that a VAR(200) (or higher) is not a reasonable choice of parametrization because of overfitting, what should I do to improve the forecast on my VAR(1) model?

One final note: If I run a lag order selection algorithm setting the maximum number of lags as 10 (I am using Python's statsmodels), I get the following output suggesting a zero-lag model (??):

VAR Order Selection (* highlights the minimums)
==================================================
       AIC         BIC         FPE         HQIC
--------------------------------------------------
0       18.30*      18.44*  8.846e+07*      18.35*
1        18.31       22.59   8.969e+07       19.94
2        18.48       26.91   1.069e+08       21.69
3        18.75       31.31   1.405e+08       23.52
4        18.99       35.69   1.818e+08       25.34
5        19.25       40.09   2.428e+08       27.17
6        19.59       44.58   3.594e+08       29.09
7        19.84       48.97   4.903e+08       30.91
8        20.21       53.48   7.732e+08       32.86
9        20.27       57.68   9.210e+08       34.49
10       20.52       62.06   1.352e+09       36.31

while if I set the maximum number of lags to 32 I get this output suggesting 32 as the optimal lag length:

  VAR Order Selection (* highlights the minimums)
==================================================
       AIC         BIC         FPE         HQIC
--------------------------------------------------
0        18.50      18.65*   1.084e+08       18.56
1        18.51       22.87   1.096e+08       20.17
2        18.69       27.27   1.321e+08       21.96
3        18.96       31.75   1.735e+08       23.82
4        19.20       36.20   2.252e+08       25.67
5        19.45       40.67   2.979e+08       27.52
6        19.79       45.23   4.424e+08       29.47
7        20.03       49.68   6.018e+08       31.32
8        20.37       54.24   9.262e+08       33.26
9        20.44       58.52   1.120e+09       34.93
10       20.68       62.97   1.648e+09       36.77
:          :           :         :             :
:          :           :         :             :
:          :           :         :             :
20       18.54       103.0   2.228e+10       50.67
21       17.87       106.5   3.225e+10       51.61
:          :           :         :             :
:          :           :         :             :
:          :           :         :             :
29      -2.060       120.3   2.226e+10       44.51
30      -8.543       118.0   1.276e+10       39.63
31      -19.91       110.9   6.047e+08       29.87
32     -36.84*       98.18  1.363e+07*      14.54*

If I try to input a higher number as maxlag the function does not converge and predictions for a VAR(32) are wildly inaccurate. I am not sure about what to make of this behaviour. Could there be something in the structure of the series that makes the model completely unable to provide meaningful predictions?

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  • $\begingroup$ You can’t predict stock prices like this $\endgroup$
    – Aksakal
    Sep 30, 2021 at 21:30
  • $\begingroup$ Could you tell how you chose the values for the intercepts and coefficients in the VAR formula? $\endgroup$
    – rain
    Oct 24, 2021 at 8:59
  • $\begingroup$ In the 1st panel of the 2nd set of plots, you can plainly see that it is not stationary as the variance changes over time. You shouldn't rely blindly on tests like ADF, especially when they are sensitive to assumptions (which are violated here). This might sound like a small detail, but it isn't: investing requires balancing risk and reward, and here you have disregarded risk (variance) and focused only on the reward (the point forecast), implicitly assuming that you could come up with a VAR model that would be "good enough" that the risk was zero. This is not a good way to invest money. $\endgroup$
    – Chris Haug
    Oct 24, 2021 at 13:00

1 Answer 1

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Could there be something in the structure of the series that makes the model completely unable to provide meaningful predictions?

Yes, indeed. Due to the nature of stock markets, price prediction with a VAR model is hopeless for daily data. The best model for price differences is VAR(0). Your VAR(200) model is certainly overfitted and it should be worse than VAR(0) for out-of-sample forecasts by any reasonable metric.

Also note that the trajectory of the best prediction does not necessarily mimic the patterns seen in historical data. E.g. if your data is i.i.d. N(0,1), the optimal prediction under a symmetric loss function is 0. If you iterate it over time, you get a straight horizontal line -- nothing like the past data. Yet such a prediction is impossible to beat (under a symmetric loss function, that is). The case of stock price increments is qualitatively similar to that.

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  • $\begingroup$ That is what I suspected. Is there any stock-related daily data that behaves differently (for instance volume or returns)? If not, is there any "classical" linear model that is able to capture these dynamics? Or should I just go with some nonlinear function like neural networks or random forests? $\endgroup$
    – Roy Hogg
    Oct 1, 2021 at 8:08
  • $\begingroup$ @RoyHogg, volume might have some patterns in it that are amenable to VAR analysis, though univariate AR might be more relevant. For stock prices, their changes or returns nothing will work. Neural networks and random forests will overfit just as much as a high-order VAR. This is the nature of the stock market. A qualitative argument is given in my last paragraph. As you may notice, the problem is not that VAR is inadequate. The problem is that in a stock market, nothing beats a straight horizontal line. $\endgroup$ Oct 1, 2021 at 8:12

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