Analysis Of Stationary Time Series (1).ppt

Nityananda Sarkar Indian Statistical Institute 203 B. T. Road, Kolkata - 700108

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A time series is a sequence of observations over time. What makes it distinguishable from other statistical analyses is the explicit recognition of the importance of the order in which the observations are made. Also, unlike many other problems where observations are independent, in time series observations are most often dependent.

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Prediction of the future based on knowledge of the past (most important). To control the process producing the series. To have a description of the salient features of the series.

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Economic planning Sales forecasting Inventory (stock) control Evaluation of alternative economic strategies Budgetting Financial risk management Model evaluation

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Old/Classical/Traditional Approach Modern/Stochastic Process Approach

Although the modern approach is a more general one, it is not that the classical approach has no importance at all in analysing time series data. As the saying goes, OLD IS GOLD.

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A Time series is a set of observations, each one being recorded at a specific time. (Annual GDP of a country, Sales figure etc)

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Classical decomposition model: a time series

is considered to consist of three components viz., Xt= mt+st+ct+t where mt, st, ct and t are trend, seasonality, cyclical and random (noise) components.

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Generally a plot speaks a thousands words.  However, plot needs to be studied carefully.  Following plots elaborate---plot does not speak so many words as well---need for mathematical modeling. 

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Cyclical Component: The oscillatory component in a time series where the period of oscillation is more than one year. One complete period is called a cycle. Length of cycle, intensity of cycle, occurrence of cycle are not fixed and unpredictable. EXAMPLE: Every economy goes through prosperity (boom) and depression.

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Assume that Trend is linear. Assume that there is no cyclical component. We will discuss two methods of trend determination (estimation). Method1: Moving average Method (MA).

Method 2: Mathematical curve fitting method.

Step 1: We calculate the average of first ‘n’ observations.

Step 2: Next we drop the first observation and include the ‘n+1th’ observation. This process is repeated until the last observation is reached.

Remarks: If ‘n’ is odd, the average of the first ‘n’ t h observations is tabulated with the n 1 time point. The average of the next ‘n’ 2 n  1  1 th observations is tabulated with the 2 time point and so on. Step 2: Next we drop the first observation and include the ‘n+1th’ observation. This process is repeated observation is reached.

until

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MONTHLY DATA

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5 Sale of warm blankets(in

4 millions)

3 12 month Moving

2 average

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Determination of ‘n’. For monthly data: n=12, 24, 36… For Quarterly data: n=4,8,…  Few data points are lost (at the beginning and at the end)  Cannot be used for forecasting.  Does not work well when the trend line is nonlinear. 

The method of moving average can successfully remove seasonal variations if the period of the moving average is equal to or a multiple of the period of the seasonal fluctuations; else the fluctuations will not be removed completely. 10 8

Sale of warm blankets(in

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millions) 5 month Moving

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average

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17 month Moving Average 1990m1 1990m6 1990m11 1991m4 1991m9 1992m2 1992m7 1992m12 1993m5 1993m10 1994m3 1994m8 1995m1 1995m6 1995m11 1996m4 1996m9 1997m2 1997m7 1997m12 1998m5 1998m10 1999m3 1999m8

0

Note: These averages are so calculated that seasonality does not get completely removed from the data.

Sale of warm blankets(in millions) Average yearly values

1990m1 1990m6 1990m11 1991m4 1991m9 1992m2 1992m7 1992m12 1993m5 1993m10 1994m3 1994m8 1995m1 1995m6 1995m11 1996m4 1996m9 1997m2 1997m7 1997m12 1998m5 1998m10 1999m3 1999m8

9 8 7 6 5 4 3 2 1 0

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Per Hector Wheat yield in Kg Observation

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time

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From the above plot, different people may fit different line based on their naked eye judgment. Which one is the best ‘in some sense’? The method is known as ‘Least Square Method’. This method assumes a particular form of trend, say, linear Trend, in the form as

Tt =a + b x t where a and b are unknown, to be estimated from the data.

ILLUSTRATION ON TREND and ITS REMOVAL-I

Unemployment level for Belgium (Jan 1982 - Aug 1994) 640,000 600,000 560,000 520,000 480,000 440,000 400,000 360,000 82

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Dependent Variable: Unemployment Level Belgium Sample: 1982M01 1994M08 Included observations: 152 Variable

Coefficient

Std. Error

t-ratio

p-value

constant

524254.3 4904.821 -124.4672 0.646678

9224.385 530.7924 8.183658 0.035619

56.83353 9.240564 -15.20923 18.15533

0.0000 0.0000 0.0000 0.0000

t (time) t2 t3

Note: Deterministic trend is removed first by the OLS method

Residuals after deterministic trend fit

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600,000 80,000

500,000

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400,000

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-80,000 82

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Residual

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Actual

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Fitted

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Detrended Unemployment level 80,000 60,000 40,000 20,000 0 -20,000 -40,000 -60,000 -80,000 82

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Dependent Variable: Detrended Unemployment Level Sample: 1982M01 1994M08 Included observations: 152 Variable

Coefficient

Std. Error

t-ratio

p value

constant @SEAS(2) @SEAS(3) @SEAS(4) @SEAS(5) @SEAS(6)

17779.69 -17535.44 -31606.49 -44510.50 -55341.97 -58891.23

1903.941 5280.581 5280.581 5280.581 5280.581 5280.581

9.338364 -3.320740 -5.985418 -8.429091 -10.48028 -11.15241

0.0000 0.0011 0.0000 0.0000 0.0000 0.0000

Note: Deterministic seasonality is removed from detrended (deterministic) data. Only significant seasonal dummies are reported.

Residuals of seasonality adjusted detrended unemployment level

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Residual

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Actual

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Dependent Variable: Unemployment Level of Belgium Sample: 1982M01 1994M08 Included observations: 152 Variable

Coefficient

Std. Error

t ratio

p-value

constant @SEAS(4) @SEAS(5) @SEAS(6)

516994.0 -36511.80 -46263.03 -48686.10

6323.958 19688.04 19688.04 19688.04

81.75166 -1.854516 -2.349803 -2.472877

0.0000 0.0657 0.0201 0.0145

Note: Instead of detrending first, deseasonalisation can as well be done in the first step. This slide shows the computation to this end.

Deseasonalized Unemployment Level (Belgium) 150,000

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-150,000 82

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Dependent Variable: Deseasonalized Unemployment Level Sample: 1982M01 1994M08 Included observations: 152 Variable

Coefficient

Std. Error

t-ratio

p-value

constant

18895.48

6359.941

2.971015

0.0035

t

4909.100

365.9657

13.41410

0.0000

t2

-125.0055

5.642390

-22.15470

0.0000

t3

0.650591

0.024558

26.49159

0.0000

Note: Detrending of the deseasonalise series is done in the second step

Residuals after a trend fit to the deseasonalized unemployment level 200,000

100,000 60,000

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Residual

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Actual

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Dependent Variable: Belgium Unemployment Level Sample: 1982M01 1994M08 Included observations: 152

Variable

Coefficient

Std. Error

t-ratio

constant t t2 t3 @SEAS(2) @SEAS(3) @SEAS(4) @SEAS(5) @SEAS(6)

544141.9 4822.603 -123.9371 0.646789 -17626.75 -31695.97 -44599.25 -55431.11 -58981.87

5868.978 326.8123 5.037261 0.021921 5332.856 5330.839 5329.683 5329.379 5329.927

92.71493 14.75649 -24.60406 29.50483 -3.305311 -5.945774 -8.368088 -10.40105 -11.06617

p-value

0.0000 0.0000 0.0000 0.0000 0.0012 0.0000 0.0000 0.0000 0.0000

Residuals of trend and seasonality fit to unemployment level

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Residual

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Second Illustration

IP INDEX (Apr1995 - Sep 2008)

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Dependent Variable: IP_INDEX Sample: 1995M04 2008M09 Included observations: 162

Variable

Coefficient

Std. Error

t-ratio

constant t t2

126.3000 0.157815 0.005034

2.114454 0.060684 0.000365

59.73170 2.600619 13.80040

p-value

0.0000 0.0102 0.0000

residuals after deterministic trend fit

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Dependent Variable: DETRENDED IP INDEX Sample: 1995M04 2008M09 Included observations: 162 Variable

Coefficient

Std. Error

t-ratio

p-value

constant @SEAS(2) @SEAS(3) @SEAS(4) @SEAS(5) @SEAS(6) @SEAS(7) @SEAS(8) @SEAS(9) @SEAS(10) @SEAS(11)

8.762640 -7.497238 8.375794 -12.23991 -10.73814 -14.98216 -13.60767 -14.40039 -12.20319 -14.10544 -11.78444

1.137970 1.971022 1.971022 1.923521 1.923521 1.923521 1.923521 1.923521 1.923521 1.971022 1.971022

7.700237 -3.803731 4.249467 -6.363286 -5.582547 -7.788925 -7.074356 -7.486477 -6.344193 -7.156408 -5.978847

0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

residuals after fitting seasonality on detrended IP Index 40 30 20 10 30

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Dependent Variable: IP_INDEX Sample: 1995M04 2008M09 Included observations: 162 Variable

Coefficient

Std. Error

t-ratio

constant t t2 @SEAS(2) @SEAS(3) @SEAS(4) @SEAS(5) @SEAS(6) @SEAS(7) @SEAS(8) @SEAS(9) @SEAS(10) @SEAS(11)

136.0743 0.125352 0.005223 -7.494443 8.379950 -12.30740 -10.80277 -15.04429 -13.66768 -14.45867 -12.26010 -14.11186 -11.78799

1.740104 0.039068 0.000235 1.979679 1.979777 1.933625 1.933489 1.933421 1.933421 1.933489 1.933625 1.979777 1.979679

78.19895 3.208549 22.23451 -3.785685 4.232775 -6.364936 -5.587188 -7.781177 -7.069173 -7.478022 -6.340474 -7.128004 -5.954494

p-value 0.0000 0.0016 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Residuals after fitting Trend and Seasonality 300

250 30 200 20 150

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Residual

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Actual

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Fitted

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Dependent Variable: IP_INDEX Sample: 1995M04 2008M09 Included observations: 162 Variable

Coefficient

Std. Error

t-ratio

constant @SEAS(1) @SEAS(2) @SEAS(3) @SEAS(4) @SEAS(5) @SEAS(6) @SEAS(7) @SEAS(8) @SEAS(9) @SEAS(10) @SEAS(11)

190.3462 0.392308 -5.838462 11.02308 -12.80330 -10.35330 -13.63901 -11.29615 -11.11044 -7.924725 -16.30000 -13.03077

13.43910 19.00576 19.00576 19.00576 18.66328 18.66328 18.66328 18.66328 18.66328 18.66328 19.00576 19.00576

14.16361 0.020642 -0.307194 0.579986 -0.686015 -0.554741 -0.730794 -0.605261 -0.595310 -0.424616 -0.857635 -0.685622

p-value 0.0000 0.9836 0.7591 0.5628 0.4938 0.5799 0.4660 0.5459 0.5525 0.6717 0.3925 0.4940

Residuals after fitting seasonality to IP Index 300

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Residual

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Actual

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Fitted

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Stationary series in obtained by removing the three deterministic components mt,st and ct, from xt either by “estimating” these components and then subtracting these from xt OR by taking differences

 xt= xt- xt-1, 2 xt=  xt- xt-1 =xt-xt-1-xt-1+xt-2 =xt-2xt-1+xt-2

Twelve period difference (to remove seasonality) of Belgium unemployment level 100,000 75,000 50,000 25,000 0 -25,000 -50,000 -75,000 -100,000 82

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First difference of the deseasonalized unemployment level 20,000 15,000 10,000 5,000 0 -5,000 -10,000 -15,000 -20,000 82

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twelve period difference(to remove seasonality) of IP index

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first difference of deseasonalized IP Index

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It is natural to assume that an observed series (x1,x2,…,xn) is a particular realisation of a stochastic process. A stochastic process is a family of random variables defined on a probability space. Specifying the complete form of the probability distribution is very ambitious, and we usually content ourselves with concentrating attention on the first and the second moments.

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However, it is impossible to make inferences based on the first and the second moments only. So we make a very important assumption called stationarity.

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A stochastic process is said to be strictly stationary if its properties are unaffected by a change of time origin; in other words, the joint probability distribution at any set of times t1,t2, …,tk must be the same as the joint probability distribution at times t1+m, t2+m, …, tk+m, where m is any arbitrary shift in the time axis. Obviously, strict stationarity is a strong condition.

A stochastic process is said to be weakly stationary (or covariance stationary) if (i) E(x1) = E(x2) = … = E(xn)=  (say) (ii) V(x1) = V(x2) = … = V(xn)= 2 (say) and (iii) Cov (xt,xt-k) = k, independent of t. Obviously, strong stationarity  weak stationarity but not vice versa. Only when {xt) are jointly normal, then both are equivalent. 

Ergodicity We also make the assumption of ergodicity which refers to one type of asymptotic independence. In words, asymptotic independence means that two realizations of a time series become even closer to independence, the further they are apart with respect to time. More formally, asymptotic independence can be defined as

| F ( X 1 ,..., X n , X 1k , X 2k ,..., X nk )

 F ( X1,..., X n ) F ( X1 k ,..., X n  k ) |  0 as

(1)

k  .

This means that the joint distribution of two subsequences of a stochastic process { X t } is equal to the product of the marginal distribution functions the more distant the two subsequences are from each other. A stationary stochastic process is ergodic if

1 n  lim   E ( X t   ) ( X t  k   )  0 n  n  k 1 

(2)

holds. This condition stated above would be satisfied if the autocovariances tend to zero with increasing k.

Sample Autocorrelations and Partial Autocorrelations While a time series plots give some idea about the nature of the underlying times series, it is not (always easy or possible) to conclude whether the time series is stationary or not. Therefore, it is useful to consider some statistics related to a time series. To that end, sample autocorrelations , ˆ k , are obtained as

ˆ k  ˆ k / ˆ 0

where ˆ k and ˆ0 are the estimates of

the autocovariance of lag k and variance , respectively, and these are given by

ˆk 

and

n



t  k 1

ˆ0 

n



t 1

( xt  x ) ( xt  k  x ) / n

( xt  x ) 2 / n

n

where x   xt / n

and n is the number of

t 1

observations in a given time series . For a series with stationary DGP, the sample autocorrelations typically die out quickly with increasing k. In contrast, the sample autocorrelations decays rather slowly for a non-stationary series. Note that sample autocorrelations are estimates of the actual (unknown) autocorrelations of the time series if the process is stationary.

Partial autocorrelations are also useful quantities that may convey useful information on the properties of the DGP of a given time series. The partial autocorrelation between X t and X t  k is the autocorrelation between X t and X t  k conditional on the intermediate variables X t 1 , X t  2 , …, X t k 1 i.e., it measures the “true” correlation between X t and X t  k after adjustments have been made for the intervening lags. Formally, the k th sample partial autocorrelation is the estimated value of  kk , which is,

obtained as the ordinary least squares (OLS) estimate of the coefficient  kk in the following autoregressive model of order k x     x  ...   x . u (3) t

k1 t 1

kk t  k

t

For stationary processes, partial autocorrelations also approach zero as k tends to infinity, and hence the sample partial autocorrelations should be small for large lags.

It may be worthwhile to state that autocorrelation functions and partial autocorrelation functions give useful information on specific properties (i.e., nature as well as order of the underlying stationary model) of a DGP other than stationarity.

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Further analysis is carried out with stationary (weak) time series only. Using correlogram analysis, as proposed by Box and Jenkins (1970), which means using the autocorrelation function (ACF) and partial autocorrelation function (PACF), we can tentatively identify a stationary time series i.e., whether it is autoregressive process of order p (AR (p)) or moving average process of order q (MA(q)) or the mixed ARMA (p,q) process.

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For an AR(p) process, the ACF is infinite in extent, but the PACF cuts off at lag  p+1. But, the picture is quite the opposite for an MA(q) process. Here, the ACF cuts off at lag q +1 onwards whereas the PACF is infinite in extent. For the mixed ARMA process, none of the criteria of ACF and PACF cuts off at some lag value – both are infinite in extent.

Turning now to calculating the acf, first calculate the autocovariances: 1 = Cov(yt, yt-1) = E[yt-E(yt)][yt-1-E(yt-1)] E(yt) = 0 and E(yt-1) = 0, so 1 = E[ytyt-1] 2 2 ( u   u   1 = E[ t 1 t 1 1 ut 2  ...) (ut 1  1ut 2  1 ut 3  ...) ] 2 3 2 = E[ 1 ut 1  1 ut 2  ...  cross  products] 2 3 2 5 2        = 1 1 1   ... =

1 2 (1  12 )

For the second autocorrelation coefficient, 2 = Cov(yt, yt-2) = E[yt-E(yt)][yt-2-E(yt-2)] Using the same rules as applied above for the lag 1 covariance 2 = E[ytyt-2] = E[ (ut  1ut 1  12ut  2  ...) (ut 2  1ut 3  12 ut 4  ...) ] = E[ 1 u t 2  1 u t 3  ...  cross  products] 12 2  14 2  ... = 2

2

4

2

= 12 2 (1  12  14  ...) =

12 2 (1  12 )

0 1 0

0 =

   1 2   2   (1  1 )  1     1 1 = 0    2   2  ( 1    1 ) 3   3 = 1

2 2 = 0



…

s =



s 1

 22

 ( 2  1 ) 2

(1   1 ) 2

 2 2   1    2   (1  1 )      12    2   2   (1  1 )   

0

so PACF helps to determine the order of AR process for a finite order stationary AR process.



Autoregressive Process of order p (AR(p)):

xt=1xt-1+  2xt-2+… + p xt-p+at Moving Average Process of order q (MA(q)): xt=at-1at-1- 2at-2-… -qat-q ARMA (p,q):

xt- 1xt-1-…- Pxt-p=at- 1at-1 -..-

qat-q. where at’s are white noise (0, 2a).

Not-stationary== Non-stationary, when distribution (parameters) changes over time. Various important examples are: Deterministic trend and Stochastic trend. 

Trend, as we all know, is the long-run smooth movement of a time series. While the idea of deterministic trend has been there for a long time, the concept of stochastic trend is relatively new.

Deterministic trend can be represented in terms of the following model: (4) xt  f (t )  ut where f (t )is any finite degree polynomial in time i.e., f (t )represents the trend, and ut is a stationary series with mean zero and variance  u2 . Nelson and Plosser (1982) called such processes as trend stationary processes (TSP).

It is obvious that trend defined in (4) is deterministic in nature and uncertainty is introduced only through the stationary component ut which is temporary in nature, and consequently, the uncertainty, as measured by variance around forecast, is also bounded even for the long-distance period.

If, on the other hand, xt is generated by the model xt    xt 1  ut (5) where ut is, as before, a stationary series with mean zero and variance  u2 , then xt is said to follow a difference stationary process (DSP) since the first differencing of xt makes the series to be stationary, and the underlying trend is then called the stochastic trend.

It is easy to find that in this case both mean and variance are functions of time. Further, unlike deterministic trend where the trend is completely predictable, stochastic trend exhibits a systematic variation though the variation is hardly predictable. As regards appropriate methods for removing trend from these two processes given in (4) and (5), note that the methods differ.

In case of deterministic trend, f(t) is estimated under a specific assumption about the trend function and then it is subtracted from xt . For stochastic trend, first differencing makes the series free of trend. However, if differencing is used for TSP, then it ensures stationarity of the series, but introduces a non-invertible moving average component involving ut .

In fact, Nelson and Kong (1981) have shown that the use of first differencing to eliminate the linear deterministic trend of a TSP series with white noise error would create a spurious first lag autocorrelation of the magnitude of -0.5 among the residuals. Also, the presence of unit root in the moving average component creates a problem for maximum likelihood estimation of the moving average parameter.

On the contrary, if the true process is DSP (with errors not exhibiting any cycle) and trend removal is done by a regression on time (treating it as a TSP), then the detrended series exhibits spurious periodicity (Nelson and Kong (1981)). It is, therefore, advisable that detrending of a time series so as to achieve stationarity, is to be done with caution.

   t  ut E (Yt )     t

Yt 

See that mean changes over time. One can apply OLS to estimate the model parameters.

Yt  Yt 1  ut

Yt  Y0  u1  ...  ut 1  ut

E (Yt )  Y0  E (u1 )  ...  E (un )  Y0  Y2  population variance of (Y0  u1  ...  ut 1  ut ) t

 population variance of (u1  ...  ut 1  ut )   u2  ...   u2   u2  t u2

This process is known as random walk.

Yt    Yt 1  ut Yt  t  Y0  u1  ...  ut 1  ut

E (Yt )  Y0  t

  t 2 Yt

2 u

This process is known as random walk with drift.

Fig 2 Trend Stationary Process

250

values

200

150

100 1

101

201

301 Time

401

Figure 1: Pure Random Walk

130

Values

120 110

100 90 1

101

201

301 Time

401

Fig 3: Random Walk with Drift 700

600 Values

500

400 300

200 100 1

101

201

301

Time

401

20

15

Random walk with drift 10

5

Stationary process 0

1

11

21

31

41

51

61

-5

-10

-15

Random walk

71

81

91





 

This is the most important question before undertaking any time series analysis. The answer requires an understanding of the modern concept of trend. Trend can be of two types. Examples of these two types of trend are given below. Deterministic trend:yt=+bt+ct2+at Stochastic trend : yt=+yt-1+at where at is a stationary process.



Note that in the latter, yt can be expressed as

yt=y0+ t+ aj

so that it also has linear trend as in the case of deterministic trend. The difference lies in the fact that the noise term i.e.,  aj is no longer a stationary process since its mean and variance are now functions of t. Hence, in this case although a systematic variation is exhibited, it is not predictable.



Underlying model:



yt=+yt-1+at i.e., yt=yt-1+at, = -1 Null hypothesis H0: =1 / =0 Alternative hypothesis H1: <1 / <0

 

Estimating equation:

yt = yt-1+ j yt-j+at



The test, called the augmented Dickey – Fuller test (ADF) which is originally due to Dickey & Fuller (1979, 1981), requires testing for the significance of  .



Main problem: The distribution of the test statistic does not follow any standard distribution under H0. The critical values have been computed by Dickey and Fuller. Note that the most general form of the estimating equation is

yt=0+ 1t+yt-1+ j yt-j+at .

The critical values are now different from the original DF critical values.It may be noted that seasonal dummies may also be included; but this would not result in further changes in the limiting distributions, and hence in the critical values.

LNEX_RATE 3.9

3.8

3.7

3.6

3.5

3.4 95

96

97

98

99

00

01

02

03

04

05

06

07

08

Null Hypothesis: LNEX_RATE has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic based on SIC, MAXLAG=13) t-ratio Augmented Dickey-Fuller test statistic

-2.237333

p-value

0.4653

Test critical values: 1% level 5% level 10% level

-4.016433 -3.438154 -3.143345

DLNEX_RATE .08 .06 .04 .02 .00 -.02 -.04 -.06 -.08 95

96

97

98

99

00

01

02

03

04

05

06

07

08

Null Hypothesis: DLNEX_RATE has a unit root Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=13) t-ratio

Augmented Dickey-Fuller test statistic

Test critical values: (* indicates significance)

-8.700974

p-value

0.0000*

1% level

-4.016433

5% level

-3.438154

10% level

-3.143345



One important drawback of this type of time series modelling is the fact that whereas in such analyses the parameters are assumed to be constant all throughout, in reality parameters may not remain constant over the entire sample period for which observations are available, especially if the span of the data is (moderately) large. The problem in its entirety is called the problem of STRUCTURAL CHANGE.

Exchange rate return (3rd JAN 2000 - 2nd JULY 2009) 4 3 2 1 0 -1 -2 -3 -4 500

1000

1500

2000

Quandt-Andrews unknown breakpoint test Null Hypothesis: No breakpoints within trimmed data Equation Sample: 3 2392 Test Sample: 362 2033 Statistic

Value

Prob.

Maximum LR F-statistic (Obs. 439) Maximum Wald F-statistic (Obs. 439)

9.282480 9.282480

0.0367 0.0367

Exp LR F-statistic Exp Wald F-statistic

2.172070 2.172070

0.0427 0.0427

Ave LR F-statistic Ave Wald F-statistic

2.748680 2.748680

0.0556 0.0556



In the context of unit root tests also, the implicit assumption is that the deterministic trend is correctly specified. Perron (1989) argued that if there is a break in the deterministic trend, the unit root test will lead to a misleading conclusion that there is a unit root, when in fact, there is not.



This problem is extremely important both at the level of stationary as well as nonstationary data analysis. At the level of stationary data, it was Chow (1960) who first talked about this problem and proposed a test under the assumption of an a priori known break point and assumption of constant variances. The assumption of known break point has been severely criticized.



Now, we have Andrews’ (1993, 2003) test to take care of this problem. Further, Bai (1997) has estimated the break point(s). As regards unit root tests under structural change, Perron (1989) has proposed a modified ADF test under three different types of deterministic trend function:







I. Crash model: A one – time change in the intercept of the trend function. II. Changing growth model: This allows for a change in the slope of the trend function without any sudden change in the level at the time of the break. III. Both effects are allowed.