# History Of Arim A Forecasting Method That Uses The Best Possible Coefficients For Seasonal And Trended Data

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ARIMA
ARIMA is a forecasting method that uses the ACF and PACF function to de-seasonalize and de-trend the observed data to generate the best possible coefficients for seasonal and trended data. (P,D,Q) For seasonal data and (p,d,q) for trend data. Goal of this method is to have a constant mean and constant variance to make the data stationary. The TSP of the data shows the major trend and seasonality in our data. In order to make the data stationary we de-trended and de-seasonal data using appropriate lags.

Trend difference creates the Fit constant or parallel to zero.

The AFC AND PACF determines which appropriate model we will choose from MA. AR, or ARMA model. For our case we will use MA model within ARIMA and we will use the ACF graph as our basis of choice. We will determine the p-values and t-values to determine the models capability in forecasting. Obtained from the observation of the ACF and PACF, we concluded our seasonal ARIMA (P,D,Q) will be (0,1,0) and Trend ARIMA(p,d,q) will be (1,2,1)

P-values are significant and the t-values being greater than 1.96 give us an idea that our model is good to generate using ARIMA model.
ARIMA Model: Revenue

Estimates at each iteration

Iteration SSE Parameters
0 3769.63 0.100 0.100
1 3432.40 0.250 0.125
2 3157.16 0.400 0.170
3 2932.68 0.550 0.218
4 2790.99 0.700 0.246
5 2775.08 0.752 0.228
6 2774.72 0.759 0.224
7 2774.70 0.761 0.223
8 2774.70 0.761 0.223

Relative change in each estimate less than 0.0010

Final Estimates of Parameters

Type Coef SECoef T P
MA 1 0.7613 0.0902 8.44 0.000; Significant
SMA 4 0.2232 0.1399 2.01 0.000; Significant

Differencing: 1 regular, 1 seasonal of order 4
Number of observation...

... middle of paper ...

...in-Watson Statistic

Durbin-Watson Statistic = 2.18107

The megaphone effect seen in the Trend Analysis Plot is a bad indicator to our model. The KB test says it is not significant and heteroscedasticity is present.

Multicolinearity
Our DW statistics is in the sweet spot; therefore, multicolinearity does not exist because all our VIF 's are less than 2.5. Even the r-squared is low and less than 50%.

Error Measures:
Fit MAPE: 0.059947
Fit RMSE:0.790802
Fore MAPE 0.073989; Avg. of7.4% error in forecast
Fore RMSE 1.27283; Avg. of 1.25 units of Standard Deviation in forecast

ACF and Trend analysis of Residuals.

The auto-correlation function denotes the model has unusual and random residual, but this is the best estimation we could come up with regression.The forecast looks in sync with the previous trend, cycle, and seasonality, if we look at the TPS plot below.

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