对T + 2 auto.arima = 0的预测

Question

I have a time series, non-stationary with diff(1).

Here the tests:

# Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
The value of the test statistic is: -5.0157 

# KPSS Unit Root / Cointegration Test #
The value of the test statistic is: 0.3134 

# Phillips-Perron Unit Root / Cointegration Test # 
The value of the test statistic is: -46.2957 

it's auto.arima is giving an MA(1) as result, with zero mean.

My problem is when i try forecast it. I get as result a single forecast to t+1, and all other are 0.

Here my data

> str(FBK)
Time-Series [1:85] from 1996 to 2017: 141488 146095 150483 156655 156849 ...


         Qtr1     Qtr2     Qtr3     Qtr4
1996 141487.8 146095.2 150483.5 156655.4
1997 156848.6 155937.2 159835.4 158977.9
1998 155368.6 158460.5 155292.1 151925.6
1999 149041.7 148199.1 147471.4 151097.5
2000 170866.3 160620.2 161279.7 165049.0
2001 174538.5 174186.8 168185.2 162310.0
2002 170277.2 168867.3 173917.6 174537.9
2003 166283.4 158245.0 155709.1 165411.8
2004 169761.7 178038.7 185613.5 181901.6
2005 180188.3 181989.6 182036.7 184795.6
2006 189160.3 192084.7 195370.6 204006.3
2007 210459.8 218289.7 226702.1 235539.3
2008 246431.9 257188.7 279232.2 258613.8
2009 236324.7 247540.1 269437.9 292023.5
2010 298190.5 306936.2 321430.3 322751.4
2011 326759.5 333299.8 334288.1 335262.3
2012 341727.2 344935.4 350190.1 354053.4
2013 355690.5 369544.0 371155.2 368577.7
2014 367707.9 357894.1 348534.6 349160.7
2015 338495.1 315932.2 304850.4 284496.2
2016 276963.9 273664.7 263458.5 260517.6
2017 253197.7

I'm using this code:

FBK_arima <- auto.arima(diff(FBK))

Series: diff(FBK) 
ARIMA(0,0,1)           with zero mean     

Coefficients:
         ma1
      0.4631
s.e.  0.0981

sigma^2 estimated as 65384314:  log likelihood=-874.63
AIC=1753.26   AICc=1753.41   BIC=1758.13

and when try forecast, i have this:

forecast(FBK_arima, n = 6)

        Point Forecast     Lo 80     Hi 80     Lo 95    Hi 95
2017 Q2      -3595.554 -13958.25  6767.145 -19443.93 12252.83
2017 Q3          0.000 -11420.06 11420.056 -17465.47 17465.47
2017 Q4          0.000 -11420.06 11420.056 -17465.47 17465.47
2018 Q1          0.000 -11420.06 11420.056 -17465.47 17465.47
2018 Q2          0.000 -11420.06 11420.056 -17465.47 17465.47
2018 Q3          0.000 -11420.06 11420.056 -17465.47 17465.47
2018 Q4          0.000 -11420.06 11420.056 -17465.47 17465.47
2019 Q1          0.000 -11420.06 11420.056 -17465.47 17465.47

Do someone already got something like this? Where may be the problem? Data? model? In graph seems the stationarity was not solve with diff, but i'm not sure if this is the main problem on forecasting

Answer 1

This is exactly what you should expect for any MA(1) model. That is, your model is

x_t = e_t + theta e_{t-1},

so the t+1 forecast will be

E[x_{t+1}] = E[e_t] + E[theta e_{t-1}]

E[x_{t+1}] = 0 + theta E[e_{t-1}]

E[x_{t+1}] = 0 + theta (x_t - e_t) = theta (x_t - e_t).

The forecast at t+2 is then

E[x_{t+2}] = E[e_{t+1}] + E[theta e_t}]

E[x_{t+2}] = 0 + theta E[e_t]

E[x_{t+2}] = 0 + theta 0 = 0.

And so on for lags > 2.