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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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