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Why is arima in R one time step off?

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}

3

1

$begingroup$

I've recently noticed an odd behavior in a few timeseries methods. Let's fit an arima model (ar1) to the annual subspots data

library(forecast)

library(ggplot2)



mod_arima <- arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

Now, if we use forecast to get the fit on the model, it's a year off. Compare these two plots:

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

  geom_line(y = fitted(mod_arima), 

            alpha = 0.5, lwd = 2, color = "blue")

To one where we delete the first value and tack on an NA at the end

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

   geom_line(y = c(fitted(mod_arima)[-1], NA), 

             alpha = 0.5, lwd = 2, color = "blue")

The second lines up perfectly, while the first is obviously one year off. What's going on here?

r time-series forecasting arima

share|cite|improve this question

asked 11 hours ago

jebyrnesjebyrnes

593415

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  This is completely normal if the best prediction of $y_{t+1}$ is roughly $y_{t}$, which happens when the time series is a martingale difference sequence (typical e.g. for prices of shares and other financial instruments).
  $endgroup$
  – Richard Hardy
  11 hours ago
  

  
  
  
  

add a comment | 

3

1

$begingroup$

I've recently noticed an odd behavior in a few timeseries methods. Let's fit an arima model (ar1) to the annual subspots data

library(forecast)

library(ggplot2)



mod_arima <- arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

Now, if we use forecast to get the fit on the model, it's a year off. Compare these two plots:

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

  geom_line(y = fitted(mod_arima), 

            alpha = 0.5, lwd = 2, color = "blue")

To one where we delete the first value and tack on an NA at the end

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

   geom_line(y = c(fitted(mod_arima)[-1], NA), 

             alpha = 0.5, lwd = 2, color = "blue")

The second lines up perfectly, while the first is obviously one year off. What's going on here?

r time-series forecasting arima

share|cite|improve this question

asked 11 hours ago

jebyrnesjebyrnes

593415

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  This is completely normal if the best prediction of $y_{t+1}$ is roughly $y_{t}$, which happens when the time series is a martingale difference sequence (typical e.g. for prices of shares and other financial instruments).
  $endgroup$
  – Richard Hardy
  11 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I've recently noticed an odd behavior in a few timeseries methods. Let's fit an arima model (ar1) to the annual subspots data

library(forecast)

library(ggplot2)



mod_arima <- arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

Now, if we use forecast to get the fit on the model, it's a year off. Compare these two plots:

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

  geom_line(y = fitted(mod_arima), 

            alpha = 0.5, lwd = 2, color = "blue")

To one where we delete the first value and tack on an NA at the end

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

   geom_line(y = c(fitted(mod_arima)[-1], NA), 

             alpha = 0.5, lwd = 2, color = "blue")

The second lines up perfectly, while the first is obviously one year off. What's going on here?

r time-series forecasting arima

share|cite|improve this question

asked 11 hours ago

jebyrnesjebyrnes

593415

$endgroup$

library(forecast)

library(ggplot2)



mod_arima <- arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

  geom_line(y = fitted(mod_arima), 

            alpha = 0.5, lwd = 2, color = "blue")

ggplot(sun_dat,

       aes(x = years, y = sunspots)) +

  geom_line() +

   geom_line(y = c(fitted(mod_arima)[-1], NA), 

             alpha = 0.5, lwd = 2, color = "blue")

share|cite|improve this question

asked 11 hours ago

jebyrnesjebyrnes

593415

share|cite|improve this question

asked 11 hours ago

jebyrnesjebyrnes

593415

asked 11 hours ago

jebyrnesjebyrnes

593415

asked 11 hours ago

jebyrnesjebyrnes

593415

2 Answers
2

active

oldest

votes

4

$begingroup$

As Richard Hardy writes: if your prediction $hat{y}_{t+1}$ of $y_{t+1}$ is pretty much your last observation $y_t$, then of course you would expect $hat{y}_{t+1}$ to line up with $y_t$, which would show exactly as the one year lag you wonder about.

And if you specify

arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

then you fitted exactly that: an AR(1) model. The AR(1) coefficient is estimated to be about 0.81. (With an intercept. Also, if you add the year as a regressor, you will model a trend. Did you intend to do this?)

Incidentally, if you allow auto.arima() to fit a model, it will choose an ARIMA(2,1,3) model, which will not exhibit this lag:

library(forecast)

model <- auto.arima(sunspot.year)

plot(sunspot.year)

lines(model$fit,col="red")

You could also include the known sunspot period of length 11, though auto.arima() won't automatically fit a SARIMA.

share|cite|improve this answer

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

$endgroup$

add a comment | 

3

$begingroup$

You're fitting an $ARIMA(1,0,0)$ model to your data, which means that your fitted model has the form:

$hat{Y}_{t+1}-m = a(Y_t-m) + epsilon$

So it looks like it's a year off, because all the model is doing is copying the value from the current year $Y_t$, with an adjustment $a$ and a drift term, and making that the prediction for the next year $hat{Y}_{t+1}$.

Your data looks highly cyclical, you might want to try fitting a seasonal ARIMA model instead of a simple AR(1) or AR(2).

share|cite|improve this answer

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

$endgroup$

add a comment | 

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4

$begingroup$

As Richard Hardy writes: if your prediction $hat{y}_{t+1}$ of $y_{t+1}$ is pretty much your last observation $y_t$, then of course you would expect $hat{y}_{t+1}$ to line up with $y_t$, which would show exactly as the one year lag you wonder about.

And if you specify

arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

then you fitted exactly that: an AR(1) model. The AR(1) coefficient is estimated to be about 0.81. (With an intercept. Also, if you add the year as a regressor, you will model a trend. Did you intend to do this?)

Incidentally, if you allow auto.arima() to fit a model, it will choose an ARIMA(2,1,3) model, which will not exhibit this lag:

library(forecast)

model <- auto.arima(sunspot.year)

plot(sunspot.year)

lines(model$fit,col="red")

You could also include the known sunspot period of length 11, though auto.arima() won't automatically fit a SARIMA.

share|cite|improve this answer

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

$endgroup$

add a comment | 

4

$begingroup$

As Richard Hardy writes: if your prediction $hat{y}_{t+1}$ of $y_{t+1}$ is pretty much your last observation $y_t$, then of course you would expect $hat{y}_{t+1}$ to line up with $y_t$, which would show exactly as the one year lag you wonder about.

And if you specify

arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

then you fitted exactly that: an AR(1) model. The AR(1) coefficient is estimated to be about 0.81. (With an intercept. Also, if you add the year as a regressor, you will model a trend. Did you intend to do this?)

Incidentally, if you allow auto.arima() to fit a model, it will choose an ARIMA(2,1,3) model, which will not exhibit this lag:

library(forecast)

model <- auto.arima(sunspot.year)

plot(sunspot.year)

lines(model$fit,col="red")

You could also include the known sunspot period of length 11, though auto.arima() won't automatically fit a SARIMA.

share|cite|improve this answer

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

$endgroup$

add a comment | 

$begingroup$

As Richard Hardy writes: if your prediction $hat{y}_{t+1}$ of $y_{t+1}$ is pretty much your last observation $y_t$, then of course you would expect $hat{y}_{t+1}$ to line up with $y_t$, which would show exactly as the one year lag you wonder about.

And if you specify

arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

then you fitted exactly that: an AR(1) model. The AR(1) coefficient is estimated to be about 0.81. (With an intercept. Also, if you add the year as a regressor, you will model a trend. Did you intend to do this?)

Incidentally, if you allow auto.arima() to fit a model, it will choose an ARIMA(2,1,3) model, which will not exhibit this lag:

library(forecast)

model <- auto.arima(sunspot.year)

plot(sunspot.year)

lines(model$fit,col="red")

You could also include the known sunspot period of length 11, though auto.arima() won't automatically fit a SARIMA.

share|cite|improve this answer

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

$endgroup$

arima(sunspot.year, c(1, 0, 0), xreg = 1700:1988)

library(forecast)

model <- auto.arima(sunspot.year)

plot(sunspot.year)

lines(model$fit,col="red")

share|cite|improve this answer

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

answered 10 hours ago

Stephan KolassaStephan Kolassa

48.3k8102181

3

$begingroup$

You're fitting an $ARIMA(1,0,0)$ model to your data, which means that your fitted model has the form:

$hat{Y}_{t+1}-m = a(Y_t-m) + epsilon$

So it looks like it's a year off, because all the model is doing is copying the value from the current year $Y_t$, with an adjustment $a$ and a drift term, and making that the prediction for the next year $hat{Y}_{t+1}$.

Your data looks highly cyclical, you might want to try fitting a seasonal ARIMA model instead of a simple AR(1) or AR(2).

share|cite|improve this answer

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

$endgroup$

add a comment | 

3

$begingroup$

You're fitting an $ARIMA(1,0,0)$ model to your data, which means that your fitted model has the form:

$hat{Y}_{t+1}-m = a(Y_t-m) + epsilon$

So it looks like it's a year off, because all the model is doing is copying the value from the current year $Y_t$, with an adjustment $a$ and a drift term, and making that the prediction for the next year $hat{Y}_{t+1}$.

Your data looks highly cyclical, you might want to try fitting a seasonal ARIMA model instead of a simple AR(1) or AR(2).

share|cite|improve this answer

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

$endgroup$

add a comment | 

$begingroup$

You're fitting an $ARIMA(1,0,0)$ model to your data, which means that your fitted model has the form:

$hat{Y}_{t+1}-m = a(Y_t-m) + epsilon$

So it looks like it's a year off, because all the model is doing is copying the value from the current year $Y_t$, with an adjustment $a$ and a drift term, and making that the prediction for the next year $hat{Y}_{t+1}$.

Your data looks highly cyclical, you might want to try fitting a seasonal ARIMA model instead of a simple AR(1) or AR(2).

share|cite|improve this answer

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

$endgroup$

share|cite|improve this answer

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

edited 9 hours ago

Stephan Kolassa

48.3k8102181

edited 9 hours ago

Stephan Kolassa

48.3k8102181

answered 10 hours ago

Skander H.Skander H.

3,9501233

answered 10 hours ago

Skander H.Skander H.

3,9501233