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To compare with the output from Stata, we could calculate the mean:. Note : these values are slightly different from the values in the Stata documentation because the optimizer in Statsmodels has found parameters here that yield a higher likelihood. Nonetheless, they are very close. This model is an extension of that from example 1. Here the data is assumed to follow the process:.

The new part of this model is that there is allowed to be a annual seasonal effect it is annual even though the periodicity is 4 because the dataset is quarterly. The second difference is that this model uses the log of the data rather than the level. From the first two graphs, we note that the original time series does not appear to be stationary, whereas the first-difference does.

This supports either estimating an ARMA model on the first-difference of the data, or estimating an ARIMA model with 1 order of integration recall that we are taking the latter approach. The last two graphs support the use of an ARMA 1,1,1 model. To understand how to specify this model in Statsmodels, first recall that from example 1 we used the following code to specify the ARIMA 1,1,1 model:.

The integration order must be an integer for example, here we assumed one order of integration, so it was specified as 1. In a pure ARMA model where the underlying data is already stationary, it would be 0. For the AR specification and MA specification components, there are two possiblities. The first is to specify the maximum degree of the corresponding lag polynomial, in which case the component is an integer. When the specification parameter is given as a maximum degree of the lag polynomial, it implies that all polynomial terms up to that degree are included.

What we want is a polynomial that has terms for the 1st and 4th degrees, but leaves out the 2nd and 3rd terms. To do that, we need to provide a tuple for the specifiation parameter, where the tuple describes the lag polynomial itself. In particular, here we would want to use:. In the previous example, we included a seasonal effect in an additive way, meaning that we added a term allowing the process to depend on the 4th MA lag.

It may be instead that we want to model a seasonal effect in a multiplicative way. The data process can be written generically as:. This emphasizes that just as in the simple case, after we take differences here both non-seasonal and seasonal to make the data stationary, the resulting model is just an ARMA model. The data process can be written in the form above as:. It may still be confusing to see the two lag polynomials in front of the time-series variable, but notice that we can multiply the lag polynomials together to get the following model:.

This is similar to the additively seasonal model from example 2, but the coefficients in front of the autoregressive lags are actually combinations of the underlying seasonal and non-seasonal parameters. The seasonal AR and MA specifications, as before, can be expressed as a maximum polynomial degree or as the lag polynomial itself. Seasonal periodicity is an integer.

This implies that a number of initial periods are lost to the differencing process, however it may be necessary either to compare results to other packages e. Stata's arima always uses simple differencing or if the seasonal periodicity is large. Notice that the first equation is just a linear regression, and the second equation just describes the process followed by the error component as SARIMA as was described in example 3.

One reason for this specification is that the estimated parameters have their natural interpretations. This specification nests many simpler specifications. For example, regression with AR 2 errors is:. The model considered in this example is regression with ARMA 1,1 errors. The process is then written:. First, using the model from example, we estimate the parameters using data that excludes the last few observations this is a little artificial as an example, but it allows considering performance of out-of-sample forecasting and facilitates comparison to Stata's documentation.

Next, we want to get results for the full dataset but using the estimated parameters on a subset of the data. The predict command is first applied here to get in-sample predictions. With no other arguments, predict returns the one-step-ahead in-sample predictions for the entire sample. We can also get dynamic predictions. One-step-ahead prediction uses the true values of the endogenous values at each step to predict the next in-sample value.

The process of determining the values of p, d, and q that are best for a given time series will be discussed in later sections of the notes whose links are at the top of this page , but a preview of some of the types of nonseasonal ARIMA models that are commonly encountered is given below.

The forecasting equation in this case is. If the mean of Y is zero, then the constant term would not be included. Depending on the signs and magnitudes of the coefficients, an ARIMA 2,0,0 model could describe a system whose mean reversion takes place in a sinusoidally oscillating fashion, like the motion of a mass on a spring that is subjected to random shocks.

The prediction equation for this model can be written as:. This model could be fitted as a no-intercept regression model in which the first difference of Y is the dependent variable. Since it includes only a nonseasonal difference and a constant term, it is classified as an "ARIMA 0,1,0 model with constant.

This would yield the following prediction equation:. This is a first-order autoregressive model with one order of nonseasonal differencing and a constant term--i. Recall that for some nonstationary time series e. In other words, rather than taking the most recent observation as the forecast of the next observation, it is better to use an average of the last few observations in order to filter out the noise and more accurately estimate the local mean.

The simple exponential smoothing model uses an exponentially weighted moving average of past values to achieve this effect. This means that you can fit a simple exponential smoothing by specifying it as an ARIMA 0,1,1 model without constant, and the estimated MA 1 coefficient corresponds to 1-minus-alpha in the SES formula.

In the previous two models discussed above, the problem of autocorrelated errors in a random walk model was fixed in two different ways: by adding a lagged value of the differenced series to the equation or adding a lagged value of the forecast error.

Which approach is best? A rule-of-thumb for this situation, which will be discussed in more detail later on, is that positive autocorrelation is usually best treated by adding an AR term to the model and negative autocorrelation is usually best treated by adding an MA term. In business and economic time series, negative autocorrelation often arises as an artifact of differencing. In general, differencing reduces positive autocorrelation and may even cause a switch from positive to negative autocorrelation.

First of all, the estimated MA 1 coefficient is allowed to be negative : this corresponds to a smoothing factor larger than 1 in an SES model, which is usually not allowed by the SES model-fitting procedure. Second, you have the option of including a constant term in the ARIMA model if you wish, in order to estimate an average non-zero trend.

The one-period-ahead forecasts from this model are qualitatively similar to those of the SES model, except that the trajectory of the long-term forecasts is typically a sloping line whose slope is equal to mu rather than a horizontal line. The second difference of a series Y is not simply the difference between Y and itself lagged by two periods, but rather it is the first difference of the first difference --i.

A second difference of a discrete function is analogous to a second derivative of a continuous function: it measures the "acceleration" or "curvature" in the function at a given point in time. The ARIMA 0,2,2 model without constant predicts that the second difference of the series equals a linear function of the last two forecast errors:.

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