Assignment - Air Rpm, Rail Pm

Assignment No – 1

“The work contained and presented here is my work and my work alone.”

1. For each, the time series (Air RPM, Rail PM, VMT), determine if the series have trend, seasonality, or both.

Diagnosing Trend in Air RPM: -

1. Air RPM time series plot displays a trend and is trending upwards.

[pic 1]

1. It has highly significant ACF, PACF, and IACF at lag 1
2. ACF with many significant lags decaying slowly from lag 1

[pic 2]

1. ACF with few significant values after first differencing is applied

[pic 3]

1. Unit root tests that are not significant but become significant when a first difference is applied.

[pic 4]

After First Difference: -

[pic 5]

Diagnosing Seasonality in Air RPM: -

1. Time series plot has repetitive behavior every S time units (here S=12 months)

[pic 6]

1. Significant ACF, PACF, and IACF values at lag S (here S=12 months)
2. ACF with significant values at lags that are multiples of S  (here S=12 months)

[pic 7]

1. ACF without a significant value at lag S after a difference of order S has been applied

[pic 8]

1. Seasonal unit root tests that are not significant but become significant when a difference of order S is applied.

[pic 9]

After First Seasonal Difference: -

[pic 10]

After analyzing both Trend and Seasonality in the Air RPM time series plot we can conclude that it has both Trend and Seasonality.

Diagnosing Trend in Rail PM: -

1. Rail PM time series plot displays a trend and is trending upwards.

[pic 11]

1. It has highly significant ACF, PACF, and IACF at lag 1
2. ACF with many significant lags decaying slowly from lag 1

[pic 12]

1. ACF with few significant values after first differencing is applied

[pic 13]

1. Unit root tests that are not significant but become significant when a first difference is applied.

[pic 14]

After First Difference: -

[pic 15]

Diagnosing Seasonality in Rail PM: -

1. Time series plot has repetitive behavior every S time units (here S=12 months)

[pic 16]

1. Significant ACF, PACF, and IACF values at lag S (here S=12 months)
2. ACF with significant values at lags that are multiples of S  (here S=12 months)

[pic 17]

1. ACF without a significant value at lag S after a difference of order S has been applied

[pic 18]

1. Seasonal unit root tests that are not significant but become significant when a difference of order S is applied.

[pic 19]

After First Seasonal Difference: -

[pic 20]

After analyzing both Trend and Seasonality in the Air RPM time series plot we can conclude that it neither has Trend nor Seasonality as it fails the Unit Root Test for Trend, Seasonal unit root test for Seasonality and its ACF has a significant value at lag S after a difference of order S has been applied.

Diagnosing Trend in VMT: -

1. VMT time series plot displays a trend and is trending upwards.

[pic 21]

1. It has highly significant ACF, PACF, and IACF at lag 1
2. ACF with many significant lags decaying slowly from lag 1

[pic 22]

1. ACF with few significant values after first differencing is applied

[pic 23]

1. Unit root tests that are not significant but become significant when a first difference is applied.

[pic 24]

After First Difference: -

[pic 25]

Diagnosing Seasonality in VMT: -

1. Time series plot has repetitive behavior every S time units (here S=12 months)

[pic 26]

1. Significant ACF, PACF, and IACF values at lag S (here S=12 months)
2. ACF with significant values at lags that are multiples of S (here S=12 months)

[pic 27]

1. ACF without a significant value at lag S after a difference of order S has been applied

[pic 28]

1. Seasonal unit root tests that are not significant but become significant when a difference of order S is applied.

[pic 29]

After First Seasonal Difference: -

[pic 30]

After analyzing both Trend and Seasonality in the VMT time series plot we can conclude that it has Trend not the Seasonality as it fails the Seasonal unit root test for Seasonality and its ACF has a significant value at lag S after a difference of order S has been applied.

1. For each, the time series (Air RPM, Rail PM, VMT) fit and evaluate at least two trend models.

Trend Models for Air RPM: -

I have evaluated two trend models for Air RPM time series i.e. Cubic Trend and Linear Trend.