Time Series Assignment

8/2/2019 Time Series Assignment

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:

Lahore School of Economics

Time Series Analysis of Real Exchange Rate of Pakistani Rupees per US Dollar

Group Members

Ahmer Zaman Khan

Umair Kiani

Armaghan Khan

Farrukh Hussnain

Dated

02/02/2012

Submitted to:

Dr. Syeda Rabab



Descriptive Statistics:

Mean1 StDev1 Minimum1 Maximum1

77.3516 10.3110 60.3978 90.1357

The data we have selected is monthly exchange rate in rupees between Pakistan Rupee

and the American Dollar. Since the standard deviation is high, this means that data is diversified

from the mean. The largest difference in the exchange rate is during April 2008, when the

exchange rate rose by 4.04 rupees. Moreover, the exchange rate has risen consistently over the

five years taken into account, which however, were mainly due to the instable political condition

and inept administering of the Pakistan government.

Time Series Plot:

The data is on the exchange rate between Pakistani Rupee and the American Dollar. The

data range is from the Feb 2007 to January 2012. We can see that the trend of the data is

increasing.

AugFeb AugFeb AugFeb AugFeb AugFeb

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P r i c e s_

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Time Series Plot of Prices_1



Trend Analysis:

This is the Trend Analysis of the data. Trend that can be identified from the data is

upward trend.


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P

r i c e s_

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MAPE 4.8286

MAD 3.6462

MSD 16.3621

Accuracy Measures

Actual

Fits

Variable

Trend Analysis Plot for Prices_1Linear Trend Model

Yt = 60.81 + 0.542*t



Moving Average:


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Length 4

Moving Average

MAPE 1.77922

MAD 1.36697

MSD 5.19865

Accuracy Measures

ActualFits

Variable

Moving Average Plot for Prices_1

Exponential Smoothing:


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A lpha 1.37647

Smoothing Constant

MAPE 0.80659

MAD 0.61456

MSD 1.08954

Accuracy Measures

Actual

Fits

Variable

Smoothing Plot for Prices_1

Single Exponential Method



Exponential smoothing and moving average are similar in that they both assume a

stationary, not trending, time series. They differ in that exponential smoothing takes into account

all past data, whereas moving average only takes into account k past data points. Technically

speaking, they also differ in that moving average requires that the past k data points be kept,

whereas exponential smoothing only needs the most recent forecast value to be kept.

Autocorrelation:

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A u t o c o r r e l a t i o n

Autocorrelation Function for Prices_1(with 5% significance limits for the autocorrelations)

From this we can infer that the trend in increasing.

Partial Autocorrelation:

The data is not stationary that can be seen through the partial and autocorrelation graphs.



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P a r t i a l A u t o c o r r e l a t i o n

Partial Autocorrelation Function for Prices_1(with 5% significance limits for the partial autocorrelations)

Differencing:

Month Prices Difference

Feb-07 60.7321 *

Mar-07 60.6927 -0.03942

Apr-07 60.7052 0.01253

May-07 60.6718 -0.03343

Jun-07 60.6256 -0.04621

Jul-07 60.3978 -0.22780

Aug-07 60.5145 0.11671

Sep-07 60.6376 0.12311

Oct-07 60.6795 0.04194

Nov-07 61.0003 0.32071

Dec-07 61.1798 0.17950

Jan-08 62.3667 1.18697



Feb-08 62.6185 0.25178

Mar-08 62.7500 0.13152

Apr-08 63.5556 0.80556

May-08 67.6009 4.04535

Jun-08 67.2563 -0.34465

Jul-08 70.5896 3.33332

Aug-08 74.2926 3.70302

Sep-08 77.1668 2.87412

Oct-08 80.4331 3.26632

Nov-08 79.9239 -0.50914

Dec-08 78.9238 -1.00018

Jan-09 79.0856 0.16185

Feb-09 79.4485 0.36290

Mar-09 80.2355 0.78701

Apr-09 80.3958 0.16023

May-09 80.5268 0.13102

Jun-09 80.9574 0.43062

Jul-09 82.0062 1.04879

Aug-09 82.7716 0.76540

Sep-09 82.8462 0.07460

Oct-09 83.2176 0.37137

Nov-09 83.4540 0.23647

Dec-09 84.0021 0.54811

Jan-10 84.5184 0.51629

Feb-10 84.8991 0.38068



Mar-10 84.3500 -0.54911

Apr-10 83.9386 -0.41143

May-10 84.3318 0.39321

Jun-10 85.2844 0.95259

Jul-10 85.5031 0.21871

Aug-10 85.6070 0.10392

Sep-10 85.7618 0.15478

Oct-10 85.9416 0.17986

Nov-10 85.5440 -0.39767

Dec-10 85.7072 0.16320

Jan-11 85.6778 -0.02936

Feb-11 85.3141 -0.36371

Mar-11 85.3380 0.02393

Apr-11 84.6278 -0.71022

May-11 85.2122 0.58441

Jun-11 85.7859 0.57366

Jul-11 86.0204 0.23452

Aug-11 86.6211 0.60067

Sep-11 87.4744 0.85336

Oct-11 86.9655 -0.50895

Nov-11 86.9316 -0.03389

Dec-11 89.3402 2.40860

Jan-12 90.1357 0.79550



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Autocorrelation Function for C2(with 5% significance limits for the autocorrelations)

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P a r t i a l A u t o c o r r e l a t i o n

Partial Autocorrelation Function for C2(with 5% significance limits for the partial autocorrelations)



Box Jenkins Model:

ARIMA Model: Prices_1

Estimates at each iteration

Iteration SSE Parameters0 1167.17 0.100 0.100 5.239

1 926.16 0.250 0.041 4.642

2 723.33 0.400 -0.023 4.072

3 552.15 0.550 -0.088 3.516

4 410.15 0.700 -0.155 2.969

5 295.79 0.850 -0.223 2.429

6 207.85 1.000 -0.291 1.894

7 145.80 1.150 -0.360 1.360

8 109.03 1.300 -0.428 0.822

9 98.16 1.418 -0.478 0.372

10 97.98 1.422 -0.475 0.304

11 97.97 1.422 -0.473 0.288

12 97.97 1.422 -0.472 0.284

13 97.97 1.422 -0.472 0.284

14 97.97 1.422 -0.472 0.283

Relative change in each estimate less than 0.0010

Final Estimates of Parameters

Type Coef SE Coef T P

AR 1 1.4217 0.1324 10.74 0.000

AR 2 -0.4722 0.1320 -3.58 0.001

Constant 0.2834 0.2140 1.32 0.192

Differencing: 0 regular, 1 seasonal of order 12

Number of observations: Original series 60, after differencing 48

Residuals: SS = 97.2769 (back forecasts excluded)MS = 2.1617 DF = 45

Modified Box-Pierce (Ljung-Box) Chi-Square statistic

Lag 12 24 36 48

Chi-Square 20.4 25.2 32.2 *

DF 9 21 33 *

P-Value 0.016 0.237 0.504 *

The AR1 model is not a good fit to the dataset because the constant is insignificant p value

(0.192) > 0.05 alpha value and the standard errors identified by the Box Pierce statistic are

significant because all p values are less than alpha value.



ARIMA Model: Difference

Estimates at each iteration

Iteration SSE Parameters

0 111.402 0.100 0.100 0.124

1 102.086 0.250 0.134 0.082

2 99.781 0.364 0.159 0.0493 99.774 0.370 0.161 0.045

4 99.774 0.370 0.161 0.045

5 99.774 0.370 0.161 0.045

Relative change in each estimate less than 0.0010

Final Estimates of Parameters

Type Coef SE Coef T P

AR 1 0.3702 0.1489 2.49 0.017

AR 2 0.1610 0.1519 1.06 0.295

Constant 0.0445 0.2200 0.20 0.841

Differencing: 0 regular, 1 seasonal of order 12

Number of observations: Original series 59, after differencing 47

Residuals: SS = 99.7585 (back forecasts excluded)

MS = 2.2672 DF = 44

Modified Box-Pierce (Ljung-Box) Chi-Square statistic

Lag 12 24 36 48

Chi-Square 20.2 25.9 32.6 *

DF 9 21 33 *

P-Value 0.017 0.209 0.486 *



Appropriate Model

Double Exponential Smoothing Method is best to forecast the values

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A lpha (level) 1.13526

Gamma (trend) 0.21093

Smoothing Constants

MAPE 0.780086

MAD 0.596162

MSD 0.979356

Accuracy Measures

Actual

Fits

Variable

Smoothing Plot for Prices_1Double Exponential Method

Comparison:

To see which model is best, compare the MAPE, MAD, and MSD of different models.MAPE MAD MSD

Trend 4.8286 3.6462 16.3621

Trend with differencing 400.022 0.699 1.068

Single Exp smoothing 0.80659 0.61456 1.08954

Double exp smoothing 0.780086 0.596162 0.979356

So we can conclude that the double exponential smoothing is the best method for forecasting.

Time Series Assignment

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