# Propose a model for this time series

I'm analyzing the time series DAX (I call Z) from the dataset EuStockMarket. Could you please verify if my analysis makes sense? Thank you so much for your help!

• First, I plot Z.

It's from the plot that Z does not have constant mean and thus is not stationary.

• Then, I plot its ACF

This plot suggests strong correlation at even big lags. This means no pattern of moving average term.

• Then I plot its PACF

This pattern indicates an autoregressive term of order 1.

• I differentiate Z to get diff1Z

The plot suggests diff1Z has constant mean of $$0$$.

• Then I plot its ACF and PACF.

It seems that diff1Z's PACF vanishes after $$23$$ lags and its ACF vanishes after $$20$$ lags. As such, I propose $$\texttt{Z} \sim \operatorname{ARMA}(23, 1, 20)$$.

Update: I have taken the log-differences of the data and plot the series as well as its ACF and PACF.

T <- diff(log(Z))
plot(T)
acf(T)
pacf(T)


Both ACF and PACF vanish after $$11$$ lags.

• The AR and MA orders you have selected are ridiculously high given the context. For a time series of financial returns (stock returns, index returns), ARMA(0,0) is the standard and ARMA(1,1) would be the largest reasonable model. In most cases, including this one, I would go with ARMA(0,0). But do what Ben suggests and work with log-returns instead. Dec 24 '20 at 7:26
• @RichardHardy I have taken the log-differences of the data. The resulted ACF and PACF vanished after $11$ lags. From this, I would propose ARMA(11, 11), but this is still faraway from your largest reasonable model ARMA(1,1). Could you please elaborate more on how to proceed? I'm still open with the possibility to use ARCH/GARCH models. Dec 24 '20 at 10:04
• Try ARMA(11,11) vs. ARMA(1,1) vs. ARMA(1,0) vs. ARMA(0,0) and see which one does best at out-of-sample forecasting. Use a rolling window approach. I expect you will find ARMA(11,11) distinctly worst and ARMA(0,0) best, with ARMA(1,0) a close competitor. ARMA(1,1) will either have nearly cancelling AR and MA roots and be a close competitor to ARMA(0,0) or have distinct roots and be a less close competitor. Financial theory simply persuades me more than ACF and PACF plots when it comes to stock index returns. Dec 24 '20 at 10:49
• Thank you so much for your help @RichardHardy. I wish you a happy Christmas with your family ^^ Dec 24 '20 at 10:50
• Happy modeling while you are at it* and then a Merry Christmas to you as well! *(I am running some simulations myself but will quit starting from lunch time which is coming soon.) Dec 24 '20 at 10:51

$$X_t = \log Z_t - \log Z_{t-1}.$$
• I have updated my question with the ACF and PACF of the log-differences of the data. Both function vanish after $11$ lags. Hence I would propose ARMA(11, 11) for this transformed data. However, @Richard Hardy said that ARMA(1,1) would be the largest reasonable model for these kind of data. I'm quite confused. Could you please elaborate on which model I should choose? Dec 24 '20 at 9:58