DOCSLIB.ORG

1 Introduction to Autocorrelation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Introduction to Autocorrelation 1 Introduction to autocorrelation Autocorrelation occurs when the Gauss-Markov assumption of uncorrelated er- ror terms is violated. Cov(t; s) 6= 0; for t 6= s Autocorrelation is often paired with heteroscedasticity because it is another way in which the variance-covariance matrix of the true error terms is different 0 2 from the Gauss-Markov assumption, E( ) = σ In. In this case we have errors that covary, i.e. if one eror is positive and large the next error is likely to be positive and large. In this case one error can give us information about another. In general the correlation between t and t−k is called autocorrelation of order k. The correlation between t and t−1 is called autocorrelation of order 1 and is usually denoted by ρ1. What does autocorrelation imply for the variance covariance ma- trix of errors? 2 2 3 E(1) E(12) :::E(1n) 0 . E( ) = 4 . 5 2 E(n1) E(n2) :::E(n) Here the off-diagonal elements are the covariances between the dif- ferent error terms and they are not zero. 2 Summary : Properties of Autocorrelation What are the causes of Autocorrelation? • Misspecification of the model : • Omitted variables • Systematic errors in measurement 1 Omission of variables: One factor that can cause autocorrelation is the omission of vari- ables. Suppose Yt is related to X2t and X3t, but we wrongfully do not include X3t in the model. Then the effect of X3t will be captured by the disturbances t. If X3t like many economic series exhibit a trend over time, then X3t depends on X3t−1;X3t−2 and so on. Similarly, t dpepnds on t−1; t−2 and so on. Misspecification of model: Suppose Yt is related to X2t with a quadratic relationship; 2 Yt = β1 + β2X2t + t but we wrongfully assume and estimate a straight line: Yt = β1 + β2X2t + t Then the error term obtained from the straight line will depend on 2 X2t. Systematic error in measurement: Suppose a company updates its inventory at a given period of time. If a systematic error occured then the cumulative invnetory stock will exhibit accumulated measurement errors. These errors will show up as an autocorrelated procedure. 3 First order autocorrelation : The simplest ans most commonly observed is the first-order auto- correlation. Consider the multiple regression model: Yt = β1 + β2X2t + β3X3t + ··· + βpXpi + t in which the current observation of the error term t is a function of the previous observation of the error term: t = ρt−1 + ut The coefficient ρ is called the first-order autocorrelation coeffi- cient and takes values from -1 to +1. It is obvious that the size of ρ will determine the strength of serial correlation. We can have three 2 different cases as described in the next slide. If ρ is zero, then we have no correlation. If ρ approaches unity. the value of the previous observation of the error becomes more important in determining thw value of the current error. In this case we have positive autocorrelation. If ρ approached -1, we have high degrees of negative autocorrelation. 4 Consequences of autocorrelation In this section we will discuss the consequences of autocorrelated errors; 3 • the impacts on the ordinary least squares (OLS) estimates. • effects on testing of hypothesis • effects on forecasting and prediction 5 Impacts on the OLS estimates OLS estimates are unbiased and consistent even if the errors are au- tocorrelated. However. the problem is with th efficiency of these es- timates. While proving the Gauss-Markov theorem that established efficiency, one of the stepsinvolve the minimization the variance of P the linear combination att. X X 2 2 XX Var att = at σ + atasCov (t; s) t6=s where the summation is over t 6= s. If Cov(t; s) 6= 0 the second term on the right-hand side does not vanish. Due to the fact that Cov (t; s) 6= 0 the best linear unbiased P estimator (BLUE) that minimizes Var( att) will not be the same as the OLS estimator. The OLS estimator is not BLUE and hence is inefficient. Note : The consequences of heteroscedasticity and autocorrela- tion is the same. If the autocorrelation in t is positive and the independent vari- 2 able Xt grows over time, then the estimated residual variance (^σ ) will be an underestimate and the value of R2 will be an overesti- mate. In general, the variance of the OLS estimates for regression coefficients will be biased. 6 Effects on tests of hypotheses The presence of autocorrelation has a serious effect on the testing of hypotheses. When the autocorrelation is positive and the inde- pendent variable Xt grows over time, estimated standard errors will 4 be smaller than the true standard error, and hence the estimated standard errors will be underestimated. • As a result of autocorrelation, { the t-statistics, which is divided by the standard error, will be overestimated { a regression coefficient that appears to be significant due to the overestimated t-statistic may not really be so. • Since the estimated variances of the parameters will be biased and inconsistent, thus the t-statistic and F-statistics are no longer valid. 7 Effects on prediction and forecast Forecasts based on OLS in the presence of autocorrelation will be unbiased, however they will be inefficient due to the inefficient es- timates of the regression coefficients. Suppose we ignore the AR(1) serial correlation and obtain OLS estimatesα ^ and β^. The OLS ^ prediction would bey ^t =α ^ + βxt. However in case of first order autocorrelation , t is predictable from ρt−1 + ut, provided ρ can be estimated byρ ^. Once we know t =ρ ^^t−1; the residual for the previous period ^t−1 is known at time t. Therefore, the AR(1) prediction will be ^ ^ y~t = β0 + β0xt +ρ ^~t−1 ^ ^ ^ ^ = β0 + β1xt +ρ ^(yi − β0 − β1xt−1) ^ ^ by making use of the fact that ^t−1 = yt−1 − β0 − β1xt−1. Thusy ~t will be more efficient than that obtained by the OLS procedure. 8 Summary : Properties of Autocorrelation If autocorrelation among the error terms in a regression model is ignored and the OLS procedure is used then; 5 • the estimates and forecasts based on them will still be unbiased and consistent. • the OLS estimates are no longer BLUE and will be inefficient. • The estimated variance of the regression coefficients will be biased and hence the test of hypothesis will be invalid. 6.
Load more

Recommended publications
  • Startips …A Resource for Survey Researchers
    Subscribe Archives StarTips …a resource for survey researchers Share this article: How to Interpret Standard Deviation and Standard Error in Survey Research Standard Deviation and Standard Error are perhaps the two least understood statistics commonly shown in data tables. The following article is intended to explain their meaning and provide additional insight on how they are used in data analysis. Both statistics are typically shown with the mean of a variable, and in a sense, they both speak about the mean. They are often referred to as the "standard deviation of the mean" and the "standard error of the mean." However, they are not interchangeable and represent very different concepts. Standard Deviation Standard Deviation (often abbreviated as "Std Dev" or "SD") provides an indication of how far the individual responses to a question vary or "deviate" from the mean. SD tells the researcher how spread out the responses are -- are they concentrated around the mean, or scattered far & wide? Did all of your respondents rate your product in the middle of your scale, or did some love it and some hate it? Let's say you've asked respondents to rate your product on a series of attributes on a 5-point scale. The mean for a group of ten respondents (labeled 'A' through 'J' below) for "good value for the money" was 3.2 with a SD of 0.4 and the mean for "product reliability" was 3.4 with a SD of 2.1. At first glance (looking at the means only) it would seem that reliability was rated higher than value.
    [Show full text]
  • Appendix F.1 SAWG SPC Appendices
    Appendix F.1 - SAWG SPC Appendices 8-8-06 Page 1 of 36 Appendix 1: Control Charts for Variables Data – classical Shewhart control chart: When plate counts provide estimates of large levels of organisms, the estimated levels (cfu/ml or cfu/g) can be considered as variables data and the classical control chart procedures can be used. Here it is assumed that the probability of a non-detect is virtually zero. For these types of microbiological data, a log base 10 transformation is used to remove the correlation between means and variances that have been observed often for these types of data and to make the distribution of the output variable used for tracking the process more symmetric than the measured count data1. There are several control charts that may be used to control variables type data. Some of these charts are: the Xi and MR, (Individual and moving range) X and R, (Average and Range), CUSUM, (Cumulative Sum) and X and s, (Average and Standard Deviation). This example includes the Xi and MR charts. The Xi chart just involves plotting the individual results over time. The MR chart involves a slightly more complicated calculation involving taking the difference between the present sample result, Xi and the previous sample result. Xi-1. Thus, the points that are plotted are: MRi = Xi – Xi-1, for values of i = 2, …, n. These charts were chosen to be shown here because they are easy to construct and are common charts used to monitor processes for which control with respect to levels of microbiological organisms is desired.
    [Show full text]
  • Problems with OLS Autocorrelation
    Problems with OLS Considering : Yi = α + βXi + ui we assume Eui = 0 2 = σ2 = σ2 E ui or var ui Euiuj = 0orcovui,uj = 0 We have seen that we have to make very specific assumptions about ui in order to get OLS estimates with the desirable properties. If these assumptions don’t hold than the OLS estimators are not necessarily BLU. We can respond to such problems by changing specification and/or changing the method of estimation. First we consider the problems that might occur and what they imply. In all of these we are basically looking at the residuals to see if they are random. ● 1. The errors are serially dependent ⇒ autocorrelation/serial correlation. 2. The error variances are not constant ⇒ heteroscedasticity 3. In multivariate analysis two or more of the independent variables are closely correlated ⇒ multicollinearity 4. The function is non-linear 5. There are problems of outliers or extreme values -but what are outliers? 6. There are problems of missing variables ⇒can lead to missing variable bias Of course these problems do not have to come separately, nor are they likely to ● Note that in terms of significance things may look OK and even the R2the regression may not look that bad. ● Really want to be able to identify a misleading regression that you may take seriously when you should not. ● The tests in Microfit cover many of the above concerns, but you should always plot the residuals and look at them. Autocorrelation This implies that taking the time series regression Yt = α + βXt + ut but in this case there is some relation between the error terms across observations.
    [Show full text]
  • The Bootstrap and Jackknife
    The Bootstrap and Jackknife Summer 2017 Summer Institutes 249 Bootstrap & Jackknife Motivation In scientific research • Interest often focuses upon the estimation of some unknown parameter, θ. The parameter θ can represent for example, mean weight of a certain strain of mice, heritability index, a genetic component of variation, a mutation rate, etc. • Two key questions need to be addressed: 1. How do we estimate θ ? 2. Given an estimator for θ , how do we estimate its precision/accuracy? • We assume Question 1 can be reasonably well specified by the researcher • Question 2, for our purposes, will be addressed via the estimation of the estimator’s standard error Summer 2017 Summer Institutes 250 What is a standard error? Suppose we want to estimate a parameter theta (eg. the mean/median/squared-log-mode) of a distribution • Our sample is random, so… • Any function of our sample is random, so... • Our estimate, theta-hat, is random! So... • If we collected a new sample, we’d get a new estimate. Same for another sample, and another... So • Our estimate has a distribution! It’s called a sampling distribution! The standard deviation of that distribution is the standard error Summer 2017 Summer Institutes 251 Bootstrap Motivation Challenges • Answering Question 2, even for relatively simple estimators (e.g., ratios and other non-linear functions of estimators) can be quite challenging • Solutions to most estimators are mathematically intractable or too complicated to develop (with or without advanced training in statistical inference) • However • Great strides in computing, particularly in the last 25 years, have made computational intensive calculations feasible.
    [Show full text]
  • Lecture 5 Significance Tests Criticisms of the NHST Publication Bias Research Planning
    Lecture 5 Significance tests Criticisms of the NHST Publication bias Research planning Theophanis Tsandilas !1 Calculating p The p value is the probability of obtaining a statistic as extreme or more extreme than the one observed if the null hypothesis was true. When data are sampled from a known distribution, an exact p can be calculated. If the distribution is unknown, it may be possible to estimate p. 2 Normal distributions If the sampling distribution of the statistic is normal, we will use the standard normal distribution z to derive the p value 3 Example An experiment studies the IQ scores of people lacking enough sleep. H0: μ = 100 and H1: μ < 100 (one-sided) or H0: μ = 100 and H1: μ = 100 (two-sided) 6 4 Example Results from a sample of 15 participants are as follows: 90, 91, 93, 100, 101, 88, 98, 100, 87, 83, 97, 105, 99, 91, 81 The mean IQ score of the above sample is M = 93.6. Is this value statistically significantly different than 100? 5 Creating the test statistic We assume that the population standard deviation is known and equal to SD = 15. Then, the standard error of the mean is: σ 15 σµˆ = = =3.88 pn p15 6 Creating the test statistic We assume that the population standard deviation is known and equal to SD = 15. Then, the standard error of the mean is: σ 15 σµˆ = = =3.88 pn p15 The test statistic tests the standardized difference between the observed mean µ ˆ = 93 . 6 and µ 0 = 100 µˆ µ 93.6 100 z = − 0 = − = 1.65 σµˆ 3.88 − The p value is the probability of getting a z statistic as or more extreme than this value (given that H0 is true) 7 Calculating the p value µˆ µ 93.6 100 z = − 0 = − = 1.65 σµˆ 3.88 − The p value is the probability of getting a z statistic as or more extreme than this value (given that H0 is true) 8 Calculating the p value To calculate the area in the distribution, we will work with the cumulative density probability function (cdf).
    [Show full text]
  • Probability Distributions and Error Bars
    Statistics and Data Analysis in MATLAB Kendrick Kay, [email protected] Lecture 1: Probability distributions and error bars 1. Exploring a simple dataset: one variable, one condition - Let's start with the simplest possible dataset. Suppose we measure a single quantity for a single condition. For example, suppose we measure the heights of male adults. What can we do with the data? - The histogram provides a useful summary of a set of data—it shows the distribution of the data. A histogram is constructed by binning values and counting the number of observations in each bin. - The mean and standard deviation are simple summaries of a set of data. They are parametric statistics, as they make implicit assumptions about the form of the data. The mean is designed to quantify the central tendency of a set of data, while the standard deviation is designed to quantify the spread of a set of data. n ∑ xi mean(x) = x = i=1 n n 2 ∑(xi − x) std(x) = i=1 n − 1 In these equations, xi is the ith data point and n is the total number of data points. - The median and interquartile range (IQR) also summarize data. They are nonparametric statistics, as they make minimal assumptions about the form of the data. The Xth percentile is the value below which X% of the data points lie. The median is the 50th percentile. The IQR is the difference between the 75th and 25th percentiles. - Mean and standard deviation are appropriate when the data are roughly Gaussian. When the data are not Gaussian (e.g.
    [Show full text]
  • Examples of Standard Error Adjustment In
    Statistical Analysis of NCES Datasets Employing a Complex Sample Design > Examples > Slide 11 of 13 Examples of Standard Error Adjustment Obtaining a Statistic Using Both SRS and Complex Survey Methods in SAS This resource document will provide you with an example of the analysis of a variable in a complex sample survey dataset using SAS. A subset of the public-use version of the Early Child Longitudinal Studies ECLS-K rounds one and two data from 1998 accompanies this example, as well as an SAS syntax file. The stratified probability design of the ECLS-K requires that researchers use statistical software programs that can incorporate multiple weights provided with the data in order to obtain accurate descriptive or inferential statistics. Research question This dataset training exercise will answer the research question “Is there a difference in mathematics achievement gain from fall to spring of kindergarten between boys and girls?” Step 1- Get the data ready for use in SAS There are two ways for you to obtain the data for this exercise. You may access a training subset of the ECLS-K Public Use File prepared specifically for this exercise by clicking here, or you may use the ECLS-K Public Use File (PUF) data that is available at http://nces.ed.gov/ecls/dataproducts.asp. If you use the training dataset, all of the variables needed for the analysis presented herein will be included in the file. If you choose to access the PUF, extract the following variables from the online data file (also referred to by NCES as an ECB or “electronic code book”): CHILDID CHILD IDENTIFICATION NUMBER C1R4MSCL C1 RC4 MATH IRT SCALE SCORE (fall) C2R4MSCL C2 RC4 MATH IRT SCALE SCORE (spring) GENDER GENDER BYCW0 BASE YEAR CHILD WEIGHT FULL SAMPLE BYCW1 through C1CW90 BASE YEAR CHILD WEIGHT REPLICATES 1 through 90 BYCWSTR BASE YEAR CHILD STRATA VARIABLE BYCWPSU BASE YEAR CHILD PRIMARY SAMPLING UNIT Export the data from this ECB to SAS format.
    [Show full text]
  • Simple Linear Regression
    The simple linear model Represents the dependent variable, yi, as a linear function of one Regression Analysis: Basic Concepts independent variable, xi, subject to a random “disturbance” or “error”, ui. yi β0 β1xi ui = + + Allin Cottrell The error term ui is assumed to have a mean value of zero, a constant variance, and to be uncorrelated with its own past values (i.e., it is “white noise”). The task of estimation is to determine regression coefficients βˆ0 and βˆ1, estimates of the unknown parameters β0 and β1 respectively. The estimated equation will have the form yˆi βˆ0 βˆ1x = + 1 OLS Picturing the residuals The basic technique for determining the coefficients βˆ0 and βˆ1 is Ordinary Least Squares (OLS). Values for the coefficients are chosen to minimize the sum of the ˆ ˆ squared estimated errors or residual sum of squares (SSR). The β0 β1x + estimated error associated with each pair of data-values (xi, yi) is yi uˆ defined as i uˆi yi yˆi yi βˆ0 βˆ1xi yˆi = − = − − We use a different symbol for this estimated error (uˆi) as opposed to the “true” disturbance or error term, (ui). These two coincide only if βˆ0 and βˆ1 happen to be exact estimates of the regression parameters α and β. The estimated errors are also known as residuals . The SSR may be written as xi 2 2 2 SSR uˆ (yi yˆi) (yi βˆ0 βˆ1xi) = i = − = − − Σ Σ Σ The residual, uˆi, is the vertical distance between the actual value of the dependent variable, yi, and the fitted value, yˆi βˆ0 βˆ1xi.
    [Show full text]
  • Basic Statistics = ∑
    Basic Statistics, Page 1 Basic Statistics Author: John M. Cimbala, Penn State University Latest revision: 26 August 2011 Introduction The purpose of this learning module is to introduce you to some of the fundamental definitions and techniques related to analyzing measurements with statistics. In all the definitions and examples discussed here, we consider a collection (sample) of measurements of a steady parameter. E.g., repeated measurements of a temperature, distance, voltage, etc. Basic Definitions for Data Analysis using Statistics First some definitions are necessary: o Population – the entire collection of measurements, not all of which will be analyzed statistically. o Sample – a subset of the population that is analyzed statistically. A sample consists of n measurements. o Statistic – a numerical attribute of the sample (e.g., mean, median, standard deviation). Suppose a population – a series of measurements (or readings) of some variable x is available. Variable x can be anything that is measurable, such as a length, time, voltage, current, resistance, etc. Consider a sample of these measurements – some portion of the population that is to be analyzed statistically. The measurements are x1, x2, x3, ..., xn, where n is the number of measurements in the sample under consideration. The following represent some of the statistics that can be calculated: 1 n Mean – the sample mean is simply the arithmetic average, as is commonly calculated, i.e., x xi , n i1 where i is one of the n measurements of the sample. o We sometimes use the notation xavg instead of x to indicate the average of all x values in the sample, especially when using Excel since overbars are difficult to add.
    [Show full text]
  • Regression Analysis
    Regression Analysis Terminology Independent (Exogenous) Variable – Our X value(s), they are variables that are used to explain our y variable. They are not linearly dependent upon other variables in our model to get their value. X1 is not a function of Y nor is it a linear function of any of the other X variables. 2 Note, this does not exclude X2=X1 as another independent variable as X2 and X1 are not linear combinations of each other. Dependent (Endogenous) Variable – Our Y value, it is the value we are trying to explain as, hypothetically, a function of the other variables. Its value is determined by or dependent upon the values of other variables. Error Term – Our ε, they are the portion of the dependent variable that is random, unexplained by any independent variable. Intercept Term – Our α, from the equation of a line, it is the y-value where the best-fit line intercepts the y-axis. It is the estimated value of the dependent variable given the independent variable(s) has(have) a value of zero. Coefficient(s) – Our β(s), this is the number in front of the independent variable(s) in the model below that relates how much a one unit change in the independent variable is estimated to change the value of the dependent variable. Standard Error – This number is an estimate of the standard deviation of the coefficient. Essentially, it measures the variability in our estimate of the coefficient. Lower standard errors lead to more confidence of the estimate because the lower the standard error, the closer to the estimated coefficient is the true coefficient.
    [Show full text]
  • 1 AKAIKE's INFORMATION CRITERIA (AIC) the General Form For
    AKAIKE’S INFORMATION CRITERIA (AIC) The general form for calculating AIC: AIC = -2*ln(likelihood) + 2*K where ln is the natural logarithm (likelihood) is the value of the likelihood K is the number of parameters in the model, e.g., consider the regression equation Growth = 10 + 5*age + 3*food + error ^ ^ ^ ^ 1 + 1 + 1 + 1 = 4 parameters AIC can also be calculated using residual sums of squares from regression: AIC = n*ln(RSS/n) + 2*K where n is the number of data points (observations) RSS is the residual sums of squares AIC requires a bias-adjustment small sample sizes. B&A rule of thumb: If ratio of n/K < 40, then use bias adjustment: AICc = -2*ln(likelihood) + 2*K + (2*K*(K+1))/(n-K-1) where variables are as defined above. Notice that as the size of the dataset, n, increases relative to the number of parameters, K, the bias adjustment term on the right becomes very, very small. Therefore, it is recommended that you always use the small sample adjustment. For example, consider 3 candidate models for the growth model above, their RSS values, and assume n = 100 samples in the data: Model K RSS AICc Food, Age 4 25 100*ln(25/100) + 2*4 + (2*4*(4 + 1))/(100 - 4 -1) = -130.21 Food 3 26 100*ln(26/100) + 2*3 + (2*3*(3 + 1))/(100- 3- 1) = -128.46 Age 3 27 100*ln(27/100) + 2*3 + (2*3*(3 + 1))/(100- 3- 1) = -124.68 1 MODEL SELECTION WITH AIC The best model is determined by examining their relative distance to the “truth”.
    [Show full text]
  • Review of Basic Statistics and the Mean Model for Forecasting
    Review of basic statistics and the simplest forecasting model: the sample mean Robert Nau Fuqua School of Business, Duke University August 2014 Most of what you need to remember about basic statistics Consider a random variable called X that is a time series (a set of observations ordered in time) consisting of the following 20 observations: 114, 126, 123, 112, 68, 116, 50, 108, 163, 79, 67, 98, 131, 83, 56, 109, 81, 61, 90, 92. 180 160 140 120 100 80 60 40 20 0 0 5 10 15 20 25 How should we forecast what will happen next? The simplest forecasting model that we might consider is the mean model,1 which assumes that the time series consists of independently and identically distributed (“i.i.d.”) values, as if each observation is randomly drawn from the same population. Under this assumption, the next value should be predicted to be equal to the historical sample mean if the goal is to minimize mean squared error. This might sound trivial, but it isn’t. If you understand the details of how this works, you are halfway to understanding linear regression. (No kidding: see section 3 of the regression notes handout.) To set the stage for using the mean model for forecasting, let’s review some of the most basic concepts of statistics. Let: X = a random variable, with its individual values denoted by x1, x2, etc. N = size of the entire population of values of X (possibly infinite)2 n = size of a finite sample of X 1 This might also be called a “constant model” or an “intercept-only regression.” 2 The term “population” does not refer to the number of distinct values of X.
    [Show full text]