12. Trends and Seasonality

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12. Trends and Seasonality

Time Series Analysis

Read Wooldridge, (2013) Chapter 10.5

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

Outline

I. Trends: Linear vs. Exponential

II Using Trending Variables in Regression III. Detrending Interpretation

IV. R‐Squared with trending y V. Seasonality

I. Trends II. Using III. Detrending IV. R-Squared V. Seasonality 2

I. Trends

• An economic time series have a common tendency of growing overtime.

Thus, some time series contain time trend.

• Models capturing trending behavior:

– (1) Linear time trend – (2) exponential trend

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat I. Trends

Linear time trend

• Write a time series { y t } as

y t = 

₀

+ 

₁

t + e t

{ e t } is an independently, identically distributed (i.i.d.) sequence of unobservable.

• What is 

₁

? Let e t = 0

y t = y t – y t-1 = 

₁

• If 

₁

> 0, then y t has an upward trend (growing overtime)

• If 

₁

< 0, then y t has a downward trend. (shrinking overtime)

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Linear time trend

• y t =  0 +  1 t + e t

• E(y t ) is linear in t.

y t =  0 +  1 t + e t E(y t ) =  0 +  1 t

• If  1 > 0, then y t has an upward trend (growing overtime)

• If  1 < 0, then y t has a downward trend. (shrinking overtime)

Usefulness of a linear time trend:

Example: Linear time trend

Let baht$ be the exchange rate (bath/$)

$ = 16.0 + .4143t s.e. (1.37) (.054) t‐stat [11.64] [7.61]

n=43 (1960‐2002), R 2 =.585749, R 2 ‐bar=.575645

Interpretation:

1) Interpret the coefficient on t

2) Does baht$ exhibit a linear time trend? (Yes/No)

Dependent Variable: BAHT$

Method: Least Squares

Included observations: 43 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 15.99318 1.374601 11.63478 0

T 0.414364 0.054421 7.614048 0

R-squared 0.585749 Mean dependent var 25.10919

Adjusted R-squared 0.575645 S.D. dependent var 6.798237 S.E. of regression 4.428544 Akaike info criterion 5.859414

Sum squared resid 804.0921 Schwarz criterion 5.94133

Log likelihood -123.977 F-statistic 57.97372

Durbin-Watson stat 0.251218 Prob(F-statistic) 0

Regress baht$ on t

I. Trends II. Using III. Detrending IV. R-Squared V. Seasonality 7

Exponential trend – in a time series

• It is captured by modeling the natural logarithm of the series as a linear trend.

log(y t ) =  0 +  1 t + e t t = 1, 2, ...

Let e t = 0

log(y t ) = log(y t ) – log(y t‐1 )

log(y t ) =  1 for all t (approximation)

• A series has the same average growth rate from period to period.

• Interpretation:  1 is an approximate average growth rate in y t .

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Usefulness of an exponential time trend

• Exponential time trend: year 1960‐2002 rgdp: real GDP, using 1988 as base year

log( ) = 5.35 + .0686t s.e. (.0276) (.0011) t‐stat (194.2) (62.83)

n=43, R 2 =.989722, R 2 ‐bar=.989471 Interpretation

1) The coefficient on time: the proportionate change in rgdp for an absolute in change in t.

2) Does RGDP exhibit an exponential trend? (Yes/No)

Dependent Variable: LOG(RGDP) Method: Least Squares

Sample(adjusted): 1960 2002

Included observations: 43 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 5.353477 0.02757 194.176 0

T 0.068583 0.001092 62.83254 0

Adjusted R-squared 0.989471 S.D. dependent var 0.865622 S.E. of regression 0.088823 Akaike info criterion -1.95895

Sum squared resid 0.323469 Schwarz criterion -1.87704

Log likelihood 44.11744 F-statistic 3947.928

Durbin-Watson stat 0.193977 Prob(F-statistic) 0

Regress log(rgdp) on t

Approximate and Exact Change Revisited:

• log( ) = 5.35 + .0686t

Approximate vs Exact percentage change Instantaneous vs Compound rate of growth.

• The coefficient on t:

The growth rate of 6.86% is the instantaneous (at a point in time) rate of growth.

• Exact percentage change =100*[exp( )‐1] =7.10 The

compound rate of growth of real GDP was about 7.48 percent per year over 1960‐2002.

A quadratic time trend

• Time Trends can be more complicated: a quadratic time trend.

y =  0 +  1 t +  2 t 2 + e t

• Interpret: Suppose  1 > 0;  2 < 0

Time t has diminishing marginal effect on output. This implies that an increasing trend is eventually followed by a decreasing trend.

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II. Using Trending Variable with Other Regressors

• Consider the model

y

_t

= 

₀

+ 

₁

x

_t1

+ 

₂

x

_t2

+ 

₃

t + u (1)



₃

> 0 implies y is growing overtime



₃

< 0 implies y is shrinking overtime

• If the model (1) is the true model, then omitting t in (1) or running the model,

y

_t

= 

₀

+ 

₁

x

_t1

+ 

₂

x

_t2

+ u (2)

yields biased estimators of 

₁

and 

₂

. The problem is called spurious regression.

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Using Trends with Other Regressors

Spurious regression:

(2) y t =  0 +  1 x t1 +  2 x t2 + u

• Suppose y t , x t1 , and x t2 are trending overtime.

• Running (2) indicates a relationship between two or more unrelated time series processes because each has a trend.

This problem is called spurious regression.

• Omit t variable: The trending factor that affects y [which is in “u” in equation (2)] may be correlated with one or more explanatory variables.

• Remedy: include time trend in equation (2)

Example : Effect of prices on housing investment

• log( ) = ‐0.550 + 1.241log(price) (s.e.) (0.043) (0.382)

{t} {12.8} {3.25}

n = 42 R

²

= 0.208 R

²

‐bar = 0.189

invpc : real per capita housing investment price : a housing price index.

• Interpretation:

1) The elasticity of per capita investment with respect to price is positive and very significant and statistically significant.

2) Both log(invpc) and log(price) exhibit upward trends

Regress log(invpc) on t; coefficient on “t” = 0.0081; t‐value = 4.5 Regress log(price) on t; coefficient on “t” = 0.0044; t‐value = 10.4

Do we have a problem of spurious regression?

Dependent Variable: LOG(INVPC)

Method: Least Squares Sample: 1 42

Included observations: 42

C -0.55024 0.043027 -12.78824 0

LOG(PRICE) 1.240943 0.382419 3.244981 0.0024

R-squared 0.20839 Mean dependent var -0.66616

Adjusted R-squared 0.188599 S.D. dependent var 0.172543

S.E. of regression 0.155423 Akaike info criterion -0.83888

Durbin-Watson stat 0.814165 Prob(F-statistic) 0.002376

Eviews: log(invpc) c log(price)

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Dependent Variable: LOG(INVPC)

C -0.84129 0.044744 -18.80234 0

T 0.008146 0.001813 4.493368 0.0001

R-squared 0.335442 Mean dependent var -0.66616

Durbin-Watson stat 1.014053 Prob(F-statistic) 0.000059

Eviews: log(invpc) c t

Dependent Variable: LOG(PRICE)

C -0.18839 0.010512 -17.92079 0

T 0.004417 0.000426 10.37139 0

Durbin-Watson stat 0.355477 Prob(F-statistic) 0

Eviews: log(price) c t

Remedy: Include a time trend

• log( ) = ‐.913 ‐.381log(price) +.0098t (se) (.136) (.679) (.0035) [t‐stat] [6.71] [.561] [2.80]

n= 42, R 2 =.341, R 2 ‐bar=.307

• Interpretation:

1) The coefficient on time “t” implies that an approximate 1%

increase in invpc from year to year on average.

2) The estimated price elasticity is negative and insignificant!

3) Which model do you preferred:

the one with trend variable versus the one with no trend variable?

Dependent Variable: LOG(INVPC) Method: Least Squares

Sample: 1 42

C -0.91306 0.135613 -6.73281 0

LOG(PRICE) -0.38096 0.678835 -0.5612 0.5779

T 0.009829 0.003512 2.798444 0.0079

Eviews: log(invpc) c log(price) t

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III. Detrending Interpretation of Regressions

• Consider the model

y t =  0 +  1 x t1 +  2 x t2 +  3 t + u t

• Detrending – is to take out the effect of “time trend” from variables (partialling‐out interpretation)

• Steps in detrending y t , x t1 , x t2

Step 1: Regress each of y t , x t1 , x t2 on a constant and the time trend t.

– For example, in detrending y t , we estimate the model y t =  0 +  1 t + e t

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat III. Detrending Interpretation

Steps in detrending y t , x t1 , x t2

Step 2: Save residuals, denoted as y t * (or ̂ ), x t1 * and x t2 * – for example, y t * = y t – + t

– In Eviews, save residuals under the names, detrend_y and detrend_x1 and detrend_x2

Step 3: Run the regression of

detrend_y on detrend_x1 and detrend_x2 (with no intercept)

Detrending Model vs. Model with time trend

• Compare to the model with time trend (t) log( ) = ‐.913 – .381log(price) +.0098t

(se) (.136) (.679) (.0035) [t‐stat] [6.71] [‐.561] [2.80]

n= 42, R 2 =.341, R 2 ‐bar=.307

• In summary, the slope coefficient, remains unchanged.

_ = ‐8.61E‐17 ‐.381detrend_lprice (s.e.) (0.021885) (.670296)

[t‐stat] [‐3.93E‐15] [‐0.568348]

n=42 R

²

=.008011 R

²

‐bar= ‐0.01679

_ = ‐.381detrend_lprice

(s.e.) (.662071) [t‐stat] [‐.57541]

n=42 R

²

=.008011 R

²

‐bar=.008011

• Model when detrending log(invpc) and log(price)

Dependent Variable: DETREND_LINVPC Method: Least Squares

Sample: 1 42

DETREND_LPRICE -0.38096 0.662071 -0.57541 0.5682

R-squared 0.008011 Mean dependent var -8.52E-17

Log likelihood 23.4593 Durbin-Watson stat 1.048727

Eviews: Regress detrend_linvpc on detrend_lprice (with no intercept)

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Dependent Variable: DETREND_LINVPC Method: Least Squares

Sample: 1 42

C -8.61E-17 0.021885 -3.93E-15 1

DETREND_LPRICE -0.38096 0.670296 -0.568348 0.573

Adjusted R-squared -0.01679 S.D. dependent var 0.140658

Eviews: Regress detrend_linvpc on detrend_lprice (with intercept)

When should you include time trend (Eg. t)

1) When y and x are trending overtime

• If adding a time trend changes the results in

important ways, the original should be treated with suspicion.

2) When y is not trending, but x is trending.

• Excluding a trend may make it look as if x has no effect on y, even though movements of x about its trend may affect y.

I. Trends II. Using III. Detrending IV. R-Squared V. Seasonality 26

IV. Computing R 2 when y is trending

• R 2 in time series regressions are often very high, compared with R 2 for cross sectional data.

Reason : When y is trending, R 2 for time series regressions can be artificially high.

• R 2 = 1 –SSR/SST= 1 ‐  u 2 / y 2

– SST/(n‐1) can substantially overestimate the true variance of y t. .

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat IV. R²when y is trending

Computing R 2 when y is trending

• Remedy : rewrite R 2 as R 2 = 1 – SSR/y t * 2

where y t * is detrended y t

• This R 2 can be found by the regression of y t * on x t1 , x t2 , t

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Effect on housing investment revisited



Original Equation and R

²

log( ) = ‐.913 – .38096log(price) +.0098t (se) (.136) (.679) (.0035) [t‐stat] [6.71] [.561] [2.80]

n= 42, R

²

=.341, R

²

‐bar=.307

 Use detrend_linvpc

_ = ‐0.0177 ‐.38096log(price) .001683t

(s.e.) (.135613) (.678835) (.003512) [t‐stat] [‐.529208] [‐.561198] [.479138]

n=42, R

²

=.008011 R

²

‐bar= ‐.04286 Note that R

^2‐

bar = 1 – [SSR/(n‐k‐1)]/[y

_t

*

²

/(n‐2)]

Dependent Variable: DETREND_LINVPC Method: Least Squares

Sample: 1 42

C -0.07177 0.135613 -0.529208 0.5997

LOG(PRICE) -0.38096 0.678835 -0.561198 0.5779

T 0.001683 0.003512 0.479138 0.6345

Adjusted R-squared -0.04286 S.D. dependent var 0.140658

Regress detrend_linvpc on log(price) t to find R 2

V. Seasonality

• For monthly or quarterly data, a time series may exhibit seasonality.

– Example: U.S. Retail sales in the fourth quarters

In most years, retailed car sales in the fourth quarter are higher than retailed sales in other quarters.

– Example: interest rate and inflation rate Not all time series data exhibit seasonality.

• Generally, most BOT data, either monthly or quarterly data, have been seasonally adjusted.

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat V. Seasonality

To Work with Seasonally Unadjusted data

• Data are seasonality unadjusted. Consider the model of apparel demand.

apparel t =  0 +  1 price t +  2 income t + u t y t =  0 +  1 x 1t +  2 x 2t + u t

• A model for quarterly data,

(**) y t =  0 +  2 QR2 t +  3 QR3 t +  4 QR4 t +  1 x t1 +  2 x t2 + u t where QR2 t , QR3 t , and QR4 t are dummy variables and the first quarter (QR1) is the base quarter

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Example: Seasonality on sales of apparel and accessory stores over the period, 1983-1986.

appar : sales of apparel and accessory stores (millions $)

appar t =  0 +  2 QR2 t +  3 QR3 t +  4 QR4 t + u t

YEAR Quarter 1 Quarter 2 Quarter 3 Quarter 4

1983 4190 4927 6843 6912

1984 4521 5522 5350 7204

1985 4902 5912 5972 7987

1986 5458 6359 6501 8607

Example:

appar : sales of apparel and accessory stores (millions $)

appar

_t

= 

₀

+ 

₂

QR2

_t

+ 

₃

QR3

_t

+ 

₄

QR4

_t

+ u

_t

= 4767.8 + 912.3QR2

_t

+ 1398.8QR3

_t

+ 2909.8QR4

_t

[prob] [0] [.0698] [.010] [0]

n=16, R

²

=.778998, R

²

‐bar=.723747 1) Interpret

2) Interpret

3) What is the seasonally adjusted sales in 1983:2?

Seasonally adjusted sales = actual value – sale difference = 4927 – 912.3 = 4014.75

YEAR Quarter 1 Quarter 2 Quarter 3 Quarter 4

1983 4190 4927 6843 6912

Dependent Variable: APPAR Method: Least Squares Sample: 1983:1 1986:4 Included observations: 16

C 4767.75 324.0365 14.71362 0

QR2 912.25 458.2569 1.990696 0.0698

QR3 1398.75 458.2569 3.052327 0.01

QR4 2909.75 458.2569 6.349605 0

Adjusted R-squared 0.723747 S.D. dependent var 1233.022

S.E. of regression 648.0731 Akaike info criterion 15.9982

Sum squared resid 5039985 Schwarz criterion 16.19135

Log likelihood -123.986 F-statistic 14.09937

Dummies and Seasonality

R 2 and Desesonalized regressand

• R 2 : If apparel t (or y t ) has pronounced seasonality, the better goodness‐of‐fit measure is an R 2 based on the deseasonalized y t .

• We can use the same technique (steps) to deseasonalize the data as to detrend the data.

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Recap of Trends and Seasonality

• Trends: Linear vs. Exponential

• Using Trending Variables in Regression

• Detrending Interpretation

• R‐Squared with trending y

• Seasonality