AR Models

[fep14]

Hi guys! While I was doing some questions of the Time-Series Analysis part, I came up with a question which I can’t fully justify de answer. The question says:

Which of the following AR models is most appropiate for a time series with annual seasonality using quarterly observations?

a) b₁x_t-1 + b₂x_t-12​ + e_t​​​​​​

b) b₀ + b₁x_t-1 + b₂x_t-4​ + e_t​​​​​

b) b₀ + b₁x_t-4 + b₂x_t-12​ + e_t​​​​​

The correct answer is B. Can you explain why is it please?

Thanks

[tickersu]

@fep14

Hi guys! While I was doing some questions of the Time-Series Analysis part, I came up with a question which I can’t fully justify de answer. The question says:

Which of the following AR models is most appropiate for a time series with annual seasonality using quarterly observations?

a) b₁x_t-1 + b₂x_t-12​ + e_t​​​​​​

b) b₀ + b₁x_t-1 + b₂x_t-4​ + e_t​​​​​

b) b₀ + b₁x_t-4 + b₂x_t-12​ + e_t​​​​​

The correct answer is B. Can you explain why is it please?

Thanks

I think the answer is C, rather than B. Answer B incorporates monthly and quarterly trends, assuming each step in time is 1 month. Answer C is correct then because it incorporates quarterly and yearly trends if each step in time is 1 month (meaning t-12 is the prior year, 12 months ago).

Where did this question and answer come from? I can say the question isn’t worded as cleanly as I would expect, either, so I may be misinterpreting some of their unclear word choice.

[snowboarder]

As previous person said, I may also misinterpreted, but my reading is that:

• a) should be erased because b0 should be included

• b) is right because “using quarterly observations” means each step in time is 3 month(quarterly)

• c) is not right (but if questions said “using monthly observations” then this would be the correct answer.)

[Harrogath]

@tickersu

@fep14

Hi guys! While I was doing some questions of the Time-Series Analysis part, I came up with a question which I can’t fully justify de answer. The question says:

Which of the following AR models is most appropiate for a time series with annual seasonality using quarterly observations?

a) b₁x_t-1 + b₂x_t-12​ + e_t​​​​​​

b) b₀ + b₁x_t-1 + b₂x_t-4​ + e_t​​​​​

b) b₀ + b₁x_t-4 + b₂x_t-12​ + e_t​​​​​

The correct answer is B. Can you explain why is it please?

Thanks

I think the answer is C, rather than B. Answer B incorporates monthly and quarterly trends, assuming each step in time is 1 month. Answer C is correct then because it incorporates quarterly and yearly trends if each step in time is 1 month (meaning t-12 is the prior year, 12 months ago).

Where did this question and answer come from? I can say the question isn’t worded as cleanly as I would expect, either, so I may be misinterpreting some of their unclear word choice.

The question wording is clear enough, it just tricked your mind.

We must pay attention to two things:

1. How the data is built: daily, weekly, monthly, quarterly, annual, etc

2. What is the suspected seasonality. The question ask for annual seasonality. There are tests for error autocorrelation that can show up to 36-lag analysis, so if we see an error pike in the chart, we can find the lag level of the suspected seasonality.

An AR model for a time series is commonly built in the simplest form :

X(t) = a + bX(t-1) + e

If this form does not work well, then we can make adjustments. The most common adjustment is seasonality. In this case, an annual seasonality in a quarterly data would be built as:

X(t) = a + bX(t-1) + cX(t-4) + e

because each quarter has 3 months and 4 x 3 = 12 (annual).

Correct answer is B.

Hope this helps

[Harrogath]

@snowboarder

As previous person said, I may also misinterpreted, but my reading is that:

• a) should be erased because b0 should be included

• b) is right because “using quarterly observations” means each step in time is 3 month(quarterly)

Yup.

@snowboarder

• c) is not right (but if questions said “using monthly observations” then this would be the correct answer.)

Not that fast, the correct model would be

X(t) = a + bX(t-1) + cbX(t-12) + e

[tickersu]

@Harrogath

The question wording is clear enough, it just tricked your mind.

We must pay attention to two things:

1. How the data is built: daily, weekly, monthly, quarterly, annual, etc

2. What is the suspected seasonality. The question ask for annual seasonality. There are tests for error autocorrelation that can show up to 36-lag analysis, so if we see an error pike in the chart, we can find the lag level of the suspected seasonality.

An AR model for a time series is commonly built in the simplest form :

X(t) = a + bX(t-1) + e

If this form does not work well, then we can make adjustments. The most common adjustment is seasonality. In this case, an annual seasonality in a quarterly data would be built as:

X(t) = a + bX(t-1) + cX(t-4) + e

because each quarter has 3 months and 4 x 3 = 12 (annual).

Correct answer is B.

Hope this helps

Good catch, as I read again my own reply, I noticed I said [assuming each step is 1 month] but the question states each step is [3 months/quarterly].

Agree then B is appropriate!

[Harrogath]