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This module provides a cursory introduction to the techniques used in the analysis of time series that is intended for advanced undergraduates.

Analysis of time-series


This module offers a brief introduction of some of the issues that arise in the analysis of time-series. Most of the topics covered are those that we attacked first by statisticians and economists. As such they do not demand the more sophisticated tools used by the more modern approaches to time-series. In spite of these shortcomings, they should give you some understanding of the issues that arise with the use of times-series in econometric analyses. One final note of explanation is necessary. These notes are designed to give you a brief introduction to how Stata handles time-series data. These notes are not a substitute for reading the Stata manual, completing a forecasting course, or reading standard texts on the rather complicated field.

Time-series analysis in Stata

Throughout this module we work with US macroeconomic data included in the MS Excel file Macro data.xls . The variables are real level of investments (RINV), real gross national product (RGNP), and real interest rate (RINTRATE). The real interest rate is approximated by the difference between the nominal interest rate and the rate of change of the price index from the previous year. The data are for the years 1963 to 1982. You can replicate the analysis done here by copying this data set into a Stata file.

The first step after entering the data set into Stata , is to declare that the data set is a time-series. The command to do this is:

. tsset year

The data set can be broken into any number of time periods including daily, weekly, monthly, quarterly, halfyearly, yearly and generic. See StataCorp [2003:119-130] for more detail on this command.

Assume that we want to estimate the following regression:

R I N V t = β 0 + β 1 R G N P t + β 2 R I N T R A T E t + ε t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaadMeacaWGobGaamOvamaaBaaaleaacaWG0baabeaakiabg2da9iabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabek7aInaaBaaaleaacaaIXaaabeaakiaadkfacaWGhbGaamOtaiaadcfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaWGsbGaamysaiaad6eacaWGubGaamOuaiaadgeacaWGubGaamyramaaBaaaleaacaWG0baabeaakiabgUcaRiabew7aLnaaBaaaleaacaWG0baabeaaaaa@54DD@

using the data set in the appendix. Figure 1 shows this regression command and the resultant output.

Stata output of an OLS regression of equation (1).
OLS estimates for Equation (1).

On the surface the estimates seem “reasonable” because the signs on the two explanatory variables are what theory predicts they should be and the parameter for real GNP is statistically different from zero. However, an examination of the residuals shown in Figure 2 suggest that the error terms might exhibit autocorrelation.

Graph of the residuals of the OLS estimation of Equation (1).
The residuals appear to be autocorrelated.

There are several issues that arise here. First, what sort of models can we use to account for autocorrelation? Second, what sorts of tests exist for detecting the existence of autocorrelation? We begin with the first of these questions by introducing the concept of first-order autocorrelation. Consider the following model:

y t = β 0 + β 1 x t + ε t . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWG0baabeaakiabg2da9iabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabek7aInaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamiDaaqabaGccaGGUaaaaa@45C1@

We say that this model exhibits first-order autocorrelation if the error terms can be written as:

ε t = ρ ε t 1 + μ t , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaOGaeyypa0JaeqyWdiNaeqyTdu2aaSbaaSqaaiaadshacqGHsislcaaIXaaabeaakiabgUcaRiabeY7aTnaaBaaaleaacaWG0baabeaakiaacYcaaaa@4484@

where μ t ~ N ( 0 , σ 2 ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaSbaaSqaaiaadshaaeqaaOGaaiOFaiaad6eadaqadaqaaiaaicdacaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiOlaaaa@4108@ Equation (3) implies that the error terms in (2) are correlated with each other. It is rather easy to show that, while the estimates of the unknown parameters are unbiased, the estimates of the standard errors are biased—downward if 1 > ρ > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg6da+iabeg8aYjabg6da+iaaicdaaaa@3B39@ and upward if 1 < ρ < 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGymaiabgYda8iabeg8aYjabgYda8iaaicdacaGGUaaaaa@3CD0@ This conclusion holds as long as the source of the autocorrelation is due to (3). If, on the other hand, the source of autocorrelation among the error terms in (2) is due to omitted explanatory variables (whose effects are absorbed in the error term), we have a potentially more serious problem. In particular, if the omitted explanatory variables are correlated with the included explanatory variables (as is often true in time-series), then the estimates of the unknown slope parameters are also biased.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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