AutocorrelationMeansRandom error termOfexpected valueThere areCorrelation, said that there is autocorrelation or sequence correlation between random error terms, which was proposed in 1972.
At this time, it is said that there are random error termsAutocorrelation(autocorrelation) or sequence correlation
Random error termAutocorrelationThe property can take many forms, the most common type of which is that there is first-order autocorrelation or first-order between random error termsAutoregressionForm, that is, the random error term is only related to its previous value:
perhaps
, we call this relationship first-order autocorrelation.
P-order autocorrelation can be expressed as:
Is satisfiedregression model Random error term of basic requirements.We call it p-order autoregression form, or the model has p-order autocorrelation
If the form (e) in Figure 1 is presented, it means
No autocorrelation
Figure 1 Autocorrelation
In linear regression modelRandom error termSequence related problems are common, especially in applicationstime seriesThe serial correlation of random error term often occurs in data processing.
Causes of autocorrelation:
There are many reasons for the serial correlation of random error terms in linear regression models, but the main reason isEconomic variablesIts own characteristics, data characteristicsvariable selection And modelFunctional formSelect the caused.
1. Autocorrelation of random error term caused by inertia of economic variable
3. Caused by interference or influence of some random factorsRandom error termAutocorrelation
4. ModelSetting errorCausing autocorrelation of random error terms
5. Correlation of random error sequence caused by observation data processing
Consequences of autocorrelation:
linear correlationWhen the random error term of the model has autocorrelation, OLS (ordinary least square method) is usedparameter estimation , will cause the following impacts.
From Gauss-MarkovTheoremProof processIt can be seen that only under the condition of the same variance and non autocorrelation, the OLS estimation has the minimum variance.When the model has autocorrelation, the OLS estimation is stillUnbiased estimation, but no longer hasEffectiveness。This and existenceHeteroscedasticityThe situation is the same as that of, indicating that there are other parameter estimation methodsEstimation errorLess than the error of OLS estimation;In other words, for the model with autocorrelation, other methods should be used to estimate the parameters in the model.
1. Autocorrelation does not affect OLSEstimatorLinear sum ofUnbiasedness, but make it ineffective
2. Autocorrelation coefficientEstimatorThere will be considerable variance
a. Use relevant metering softwareFor example, E-VIEWS inspectionresidualDistribution of.If the residual distribution has obvious and smooth linear distribution image, it indicates thatrelevanceThe possibility of existence is high.Conversely, noneRegular fluctuationLarge distributionimage displayThe correlation is weak.
For example, in the above picture, the rounder distribution on the left shows the existence of autocorrelation, while the larger fluctuation on the right shows the opposite.
Autocorrelation judgment method 1 example
b.Durbin Watson Statistics: assuming that the time series model has autocorrelation, we assume thatError termIt can be expressed as Ut=ρ * Ut-1+εThe hypothesis is set up by statistical test. If ρ=o., there is no autocorrelation.Durbin-Watsonstatistic(hereinafter referred to as DW statistics) can be used as a tool to judge positive, negative and zero (no) correlations.DW statistic: d=∑ (Ut-Ut-1) ^ 2/∑ ut ^ 2 ≈ 2 * (1 - ρ). If d=2, there is basically no selfCorrelationThere is a positive correlation when d is close to 0, and a negative correlation when d is close to 4.[1]
c. Q-Statistics uses (box-pierce）-Eviews (7th version) is an example: many statistical measurement software software provides QtestTo detect, we use Eviews as an example.The test statistics of Q is Q=n * ∑ρ ^ 2Null hypothesisNull hypothesis H0=0 has the same meaning as method 2.If the proof of null hypothesis fails, thenAntithesis hypothesisρ≠ 0 is true, which means there is autocorrelation.The Q-test in Figure 1 shows the correlation.
Figure 1 Q-test shows the correlation
How to weaken the autocorrelation of models
Method 1 (GLS or FGLS): assuming that there is an autocorrelation model, the relationship between the error terms is: Ut=ρ * Ut-i+ε (ε is the error term except autocorrelation, i.i.d~(0, σ). The model in t period is yt=β xt+Ut, and in t-1 period is ρ * yt-1=ρ * β xt-1+ρ * Ut-1.Subtract t-1 from t period to get yt-yt-1=β (xt-xt-1)+(Ut-Ut-1). It is known that Ut-Ut-1=ε.After sorting out, the new model meets Gauss Makov's assumption and the White noise condition（HomoscedasticityOr equidispersion), no autocorrelation.
Weakening method 1
Method 2 (HAC: Heteroscedasticity Autocorrelation consistent): Take Eviews as an exampleanalysis model Select HAC and gradually add the number of time lag in the model to correct DW statistics toNormal valueReduce autocorrelation.
