Diksha Mission Economics No.43 Econometrics Time Series ......Auto Regressive Integrated Moving...

Diksha Mission Economics No.43 Econometrics

Time Series, ARIMA

TIME SERIES ECONOMETRICS

The concept of time series was developed by Engel & Granger. The method tells us the arrangement of variables over a period of time.

Univariate: Analysing one sequence of data over a period of time

Example: Yt = F(Yt-1 , Yt-2, ….., Yt-n) + ut

Multi-variate: Analysing a number of variables over time)

Example: Yt = β1 + β2 Yt-1 + β3 Xt +ut

Stationary Time Series

A time series is considered to be stationary if its mean and variance do not change over time. If we explain it mathematically, the stationary time series will be that series whose:

E(Yt) = μ which is not a function of time

Var(Yt) = E(Yt - μ)2 = σ2 which is not a function of time

To join Diksha NET JRF Paper 2- Economics Preparation, Send “JOIN” to 7306721731 [WhatsApp]

Cov(Yt+k, Yt) = Ɣab(h); if a is not equal to b, then the function is known as a cross correlation function and is the equality is sustained, then autocorrelation is present in the model.

1. Strictly Stationary: It is an extreme form of stationarity. Any series ( Yt, , Yt+1…., Yt+n) is considered to be strictly stationary if the joint distribution of the first ‘n’ observations is equal to that of another set of distribution with ‘n’ observations separated by a time lag, say k (Yt+k, Yt+k+1,……., Yt+k+n). To explain it mathematically,

Assume we have two distributions (Xt1) and (Xt2).

We introduce a lag of say k in the series, (Xt1+k) and (Xt2+k).

So in a strictly stationary series, even after introducing the lag

Xt1 = Xt1+k; Xt2 = Xt2+k

Also in a strictly stationary series, the continuous variables tend to be identical to the discrete variables, that is,

X (t) ≡Xt

Where: X(t) is the continuous variable and Xt is the discrete variable.


One point to remember is that under the strictly stationary series, higher order moment will

be constant.

2. Weak Stationary: A series Yt is considered to be a weak stationary series if it’s mean and variance are independent of time, that is, in such type of a series, it’s only the mean and variance which do not change. Rest all the variables are a function of time. However, the covariance or auto covariance is the function of lag but not of time. Mathematically it can be expressed as:

E(Yt) = μ which is not a function of time

Var(Yt) = E(Yt - μ)2 = σ2 which is not a function of time

Cov(Yt+k, Yt)

E(Yt - μ)( Yt+k - μ) = Ɣk which is a function of lag but not time

Non-Stationary Series

The Non-Stationary Stochastic Process is defined as the process whose mean and variance are not constant over the period of time.


For example, if we have a stochastic process Yt. It will be a non-stationary stochastic process if;

E(Yt) = f{time(t)}

Variance (Yt) = f{time(t)}

Process generating Time Series

1. Random or Stochastic Process:

In such a process, a variable can take any value at a point of time. It is a discrete process which consists of a series of independently, identically distributed (iid) variables, also known as white noise.

We consider a process to be white noise process if it has zero mean, constant variance and is serially uncorrelated.

Mathematically it can be expressed as below:

Mean: E(Yt) = 0

Variance = E(Yt – 0)2 = σ2


2. Random Walk: It is a non-stationary process, expressed as below:

Yt = Yt-1 + ut

Where: ut is our white noise error term which is independently, identically and normally distributed with mean equal to zero and a constant variance, that is,

ut ~ iid N (0, σ2)

Current value of the endogenous variable will be embedded in the past term including the error term.

With reference to out equation above, value of Y in time period t is equal to its value in the previous time period, that is, t-1 plus a random shock/term. If we take drift and trend in the model, then it would become non-stationary.

Random walk with drift: Yt = Ϩ + Yt-1 + ut

where: Ϩ is the drift parameter

ut is the white noise error term


Random Walk with drift and trend:

Yt = Ϩ + Yt-t + βt + ut

Where: Ϩ is the drift variable,

β is the trend variable

3. Integrated Stochastic Process:

Auto-Regressive (AR) Process:

Any series is said to be generated by Auto-Regressive process if it is defined as :

Yt = βYt-1 + ut

The above mentioned model is the auto regressive model of order one, denoted as AR(1). This is so because we include one lag term of our endogenous variable Yt .

One important point to note here is that if β = 1, then the series becomes a random walk process or a unit root process as explained before.

A series is generated by auto regressive process of order p, that is, AR(p), if;

Yt = β1Yt-1 + β2Yt-2 + .......... + βp Yt-p + ut


Moving Average (MA) Process:

A series is said to be generated by moving average if it is defined as follows;

Yt = ut + Ɣ1ut-1 + Ɣ2 ut-2 + …. + Ɣq ut-q

The above mentioned model is denoted as MA(q); implying Moving Average of order q.

The difference between Auto Regressive and Moving Average Process are:


Auto Regressive Models Moving Average Models

Regression is done on the lagged values of endogenous variables

Regression is done on the lagged values of the error terms

The model is stochastic in nature, thus it has an error term

The model is deterministic in nature, therefore, it has no error terms.

Yt = β1Yt-1 + β2Yt-2 + ......... + βp Yt-p + ut Yt = ut + Ɣ1ut-1 + Ɣ2 ut-2 + …. + Ɣq ut-q

Auto Regressive Moving Average (ARMA):

A series is said to be generate by the process of Auto Regressive Moving Average process if it is defined as follows:

Yt = β1Yt-1 + β2Yt-2 + ……..+ βp Yt-p + ut + Ɣ1ut-1 + Ɣ2 ut-2 + …. + Ɣq ut-qThe m

The above mentioned series is a combination of both Auto Regressive model of order p (AR(p)) and Moving Average model of order q (MA(q)).

The general ARMA model was first described by Peter Whittle in his thesis in 1951.

He used mathematical analysis like Laurent Series and Fourier analysis with some statistical inference to explain ARMA models.

It was further popularized by George E.P.

Box and Jenkins, in their book in 1971, also introducing a method named after them (Box-Jenkins) method for choosing and estimating the ARMA models.


The process of Auto Regressive Moving Average can be summarized as

AR(p) Yt = β1Yt-1 + β2Yt-2 + ……..+ βp Yt-p + ut

MA(q) Yt = ut + Ɣ1ut-1 + Ɣ2 ut-2 + …. + Ɣq ut-q

ARMA(p,q) Yt = β1Yt-1 + β2Yt-2 + ……..+ βp Yt-p + ut +Ɣ1ut-1 + Ɣ2 ut-2 + …. + Ɣq ut-q

Auto Regressive Integrated Moving Average (ARIMA) Process:

Suppose that ΔdYt is a stationary series that can be represented by an ARMA model of order (p,q); then we can say that Yt can be represented by an ARIMA process of order (p,q, d)

The model is called integrated because the stationary ARMA model which is fitted to the difference data has to be integrated to provide a model for non-stationary data.

FUNCTIONAL FORMS

A linear functional form is of the type: yi = β1 + β2Xi + ui


However, there can be following types of functional forms as well:

1. Log-Log/ Log-Linear/ Double Log Function: Such type of a functional form is represented as:

Log yi = log β1 + β2 log Xi + ui

Here, β2 is a slope coefficient and remains constant. It tells us per unit changes in yi with per unit changes in Xi.

This model is also known as Constant Elasticity Model.

2. Log-Lin/Lin-Log Models: These are the semi log models

3. Linear Trend Models: In such models, the regression is done on a time trend and is expressed as follows:

yi = β1 + β2 t + ui

4. Reciprocal Models: Under such models, the OLS cannot be applied directly. The model is expressed as:

Yi = β1 + β2 ( 𝟏/ 𝑿𝒊 ) + ui

If Xi approaches to infinity, then yi is asymptotic to β1.


CO-INTEGRATION

The term co-integration was introduced by Engel (1987) & Granger (1981). Co-integration studies the short run and long run dynamics of the series and links the short run behavior with the long run behavior.

It is said that if there is a long run relationship between the two given series, then they are said to be co-integrated.

For example, Yt = I(1)

Xt = I(1)

Yt – βXt will become I(0), then this process is called co-integration.

If it would have been I(1); we won’t call the series to be co-integrated.

Error Correction Mechanism: It is also known as Granger Representation Theorem.

Δy = f(ΔX, u)

Where u is used to correct the disequilibrium.

If u ≠ 0, the equilibrium needs to be established and accordingly y changes to get back to equilibrium.


Error correction mechanism

This describes the short run dynamics. If Xt, Yt are co-integrated then there is a long term relationship between them. However, in the short run, there may be a possibility of disequilibrium. So the error term can be treated as an equilibrium error and this can be used to tie the short run behavior to the long run value.

Granger Representation Theorem

The theorem states that if two variables are co-integrated, then the relationship between the two can be expressed as an error correction mechanism.

Tests of Co-integration

1. Dickey-Fuller and Augmented Dickey Fuller Test: It is also called Engel-Granger Test/Augmented E-G Test.

2. Durbin-Watson Test: It is also called co-integrating Regression Durbin-Watson test.

Under this we take null hypothesis (H0: d=0).

We assume that if d = 0, then we have a unit root, hence the series is non-stationary.

If H0 is rejected, then this would imply that the series is stationary in nature and hence no problem.


VECTOR AUTO REGRESSION (VAR)

The concept was given by Sims and is related to simultaneous models and Granger Causalty. Under this, we consider a number of simultaneous equations and also that all variables are dependent. It is a multiple time series generalization of auto-regressive models.

We do not have to differentiate between endogenous and exogenous variables. Estimation becomes simple. OLS can be used making our estimators to be precise in nature. We can forecast a number of variables at a time.

We do not consider it as a purely econometric approach because this kind of approach is based on less a-priori information.

It is difficult to handle lags and estimate the number of lags to be taken in a model.


QUESTIONS FOR THE DAY


1. The basic construction of price index number involves which of the following steps

1.Selection of a base year and price of a group of commodities in that year

2.Prices of a group of commodities for the given year that are to be compared

3.Changes in the price of the given year are shown as percentage variation for the base year

4.Index number of price of a given year is denoted as100

Codes:

A.1 ,2 and3 B.2,3 and4

C.1,3 and4 D.1,2and4

2. if the distribution is skewed to the left, then it is

A. Asymmetrically skewed

B. Symmetrical

C. Negatively skewed

D. Positively skewed

To join Diksha NET JRF Paper 2- Economics Preparation, Send “JOIN” to 7306721731

[WhatsApp]

3. Consider the following measures

1.Correlation coefficient 2.Covariance

3.Coefficient of variation 4.Index number

Which of these are unit free

A. 1 and2

B. 2 and3

C. 1 and4

D. All of the above

4. In case of high income inequality ,the income distribution is

A. A symmetric distribution

B. U shaped distribution

C. Inverted J-Shaped distribution

D. None of the above


5. Consider the following statements

1.Quartile deviation is more instructive range as it discards the dispersion of extreme items

2.Coefficient of quartile deviation cannot be used to compare the degree of variation in different distributions

3.There are 10 deciles for a series

Codes:

A. 1,2 and 3 B. 2 only

C. 3 only D. 1 only

Answers:-

1) A 2) C 3) D 4) C 5) D


Diksha Mission Economics No.43 Econometrics Time Series ......Auto Regressive Integrated Moving...

Documents

Transcript of Diksha Mission Economics No.43 Econometrics Time Series ......Auto Regressive Integrated Moving...