Explain the various concepts & tools of regression with time series data?

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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Chapter4: Equations Of Linear Functions
Section4.6: Regression And Median-fit Lines
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Explain the various concepts & tools of regression with time series data?

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Step 1
A time series can be decomposed into three components: the trend (long term direction), the seasonal (systematic, calendar related movements) and the irregular (unsystematic, short term fluctuations)   
 
Time series regression is a statistical method for predicting a future response based on the response history (known as autoregressive dynamics) and the transfer of dynamics from relevant predictors.
Time series regression is commonly used for modelling and forecasting of economic, financial, and biological systems.
 
Usually the errors in time series data is usually exhibit some type of autocorrelated structure. By autocorrelation we mean that the errors are correlated with themselves at different time periods.
Some of autocorrelation in time series regression is from failure of the analyst to include one or more important predictor variables in the model.
 
Correlation over time (Serial correlation): The correlation of a series with its own lagged values is called
autocorrelation or serial correlation. 
The first autocorrelation of Statistics homework question answer, step 1, image 1is Statistics homework question answer, step 1, image 2
 
First autocovariance of Statistics homework question answer, step 1, image 3 is  Statistics homework question answer, step 1, image 4
 
 
Autoregressive (AR) model : A natural starting point for a forecasting model is to use past
values of Y (Statistics homework question answer, step 1, image 5) to forecast Statistics homework question answer, step 1, image 6
 
An autoregression is a regression model in which Yt is regressed against its own lagged values.
The number of lags used as regressors is called the order of the autoregression. 
 
In a first order autoregression, Statistics homework question answer, step 1, image 7is regressed against Statistics homework question answer, step 1, image 8
The first order autoregressive model AR(1) is
Statistics homework question answer, step 1, image 9
 
And Statistics homework question answer, step 1, image 10 order autoregression Statistics homework question answer, step 1, image 11 is regressed against  Statistics homework question answer, step 1, image 12
The Statistics homework question answer, step 1, image 13order autoregressive model AR(p) is, 
 
Statistics homework question answer, step 1, image 14
When p =0, it means that there is no auto-correlation in the series.  When p=1, it means that the series auto-correlation is till one lag.
 
 
 
 
 
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