# first differencing solve auto-correlation

what’s the typical problem of trend model and what’s the typical problem of AR model? first differencing can solve unit root but can it solve auto-correlation

Typical problem for Trend Models is Serial Correlation. If Serial Correlation is found in Trend Model, use AR model instead. Typical problem in AR model is Stationarity, which can be removed by first differencing the data. First differencing can also solve Unit Root problem, but it cannot solve auto-correlation. Auto Correlation is solved by adding more lags of the data in regression equation.

Thanks. but I’m confused about first differencing and add lag, my understanding first differencing= AR(1)? Sales(t)-Sales(t-1), is AR(4) = sales(t)-sales(t-4)? so you are saying trend model will not have auto-correlation problem, but has serial correlation? then which model has auto-correlation problem?

can explain?

Let me explain ‘first-differencing’ first. Say you have a series of data (5, 7, 10, 15, 21, 28). For AR(1) model, you will be regressing this series with its 1 lagged value. That is, the pairs will be (5, 7), (7, 10), (10, 15), (15, 21) and (21, 28). Now if, regression of this series shows unit root problem, you first difference the series to form it as (2, 3, 5, 6, 7). Then its AR(1) pairs to regress would be (2, 3), (3, 5), (5, 6) and (6, 7). Now on auto-correlation: Auto-correlation is caused in AR models, when some adjacent lagged values, that can explain predicted value of a dependent variable, are ignored. The solution is to include those lagged values as independent variables in regression. That is, to fix Auto-correlation in AR(1) model, you may have to make it an AR(2) or AR(3) or AR(4) model…till auto-correlation is removed.

Thanks, you illustrate using AR(1), is AR(2)=(5,7,10), (7,10,15)… AR(3)=(5,7,10,15), (7,10,15,21)… a bit strange? if my original equation is Sales(t)=B1*sales(t-1)+B0, how to write equation in AR(1)? how to fix auto-correlation, if I have AR(2) T value =1.97, AR(3) T value=1.99, AR(4) T value =2.4, assuming T critical is 1.96, do I need to include AR(2), AR(3), AR(4) all together? from your explanation, it seems auto-correlation and postive serial correlation are two different animal. if I use DW test, and DW=3, DL=0.5, DU=2 is given by the question, can I conclude it is positive serial correlation?

> is AR(2)=(5,7,10), (7,10,15)…AR(3)=(5,7,10,15), (7,10,15,21) ? yes, it would be like that. > if my original equation is Sales(t)=B1*sales(t-1)+B0, how to write equation in AR(1)? What you have written is the AR(1) regression equation. > how to fix auto-correlation, if I have AR(2) T value =1.97, AR(3) T value=1.99, AR(4) T value =2.4, assuming T critical is 1.96, do I need to include AR(2), AR(3), AR(4) all together? Because, 2nd 3rd and 4th lags are statistically significant, the best model to use for regression would be AR(4) model. And not AR(2), AR(3) and AR(4) all together. >from your explanation, it seems auto-correlation and postive serial correlation are two different animal. Probably, their meaning would be same, but for exam purpose just treat them differently. Detect Serial Correlation by DW test (when independent variables are not lagged variables, that is it is NOT an AR model). And detect Auto-Correlation from the ANOVA table for an AR model. > if I use DW test, and DW=3, DL=0.5, DU=2 is given by the question, can I conclude it is positive serial correlation? There is a Serial Correlation, but it is a Negative Serial Correlation. Because DW = 2(1 - r). So, for DW values greater than 2, correlation ® would be negative and for DW values less than 2, correlation would be positive.

for your answer: (1)There is a Serial Correlation, but it is a Negative Serial Correlation. Because DW = 2(1 - r). So, for DW values greater than 2, correlation ® would be negative and for DW values less than 2, correlation would be positive. This is first time I heard that, so do you mean we only need DW, no need for comparing with DL(lower bound) and DU(higher bound), I think there is a situation, inclusive judgement (2)Because, 2nd 3rd and 4th lags are statistically significant, the best model to use for regression would be AR(4) model. And not AR(2), AR(3) and AR(4) all together I also suprised, if all are significant, we should include them all, why you remove AR(2), AR93). if AR(1) T=0.05, AR92) T =0.06, AR(3)=0.07, AR(4)=3, I will include AR(4) (3), can I write AR(3) as Sales (t)=B1*Sales(t-3)+B0?

1. NO, you DO need DL and DU. I was only trying to correct you on POSITIVE Correlation you had mentioned (it would actually be NEGATIVE Correlation) and trying to explain why it would be NEGATIVE. 2. AR(4) means you have total 4 lagged independent variables in your regression. It DOES NOT mean you ONLY have 4th lagged independent variable in your regression. 3. No, AR(3) would be Sales(t) = b0 + b1*Sales(t-1) + b2*Sales(t-2) + b3*Sales(t-3)

if DW=1, DL=0.5, DU=2, then r=0.5, can we conclude it is positive correlation? when should we compare DW with DL and Du if 1st, 2nd 3rd AR are not significant and 4th lags is statistically significant, what should we do?

> if DW=1, DL=0.5, DU=2, then r=0.5, can we conclude it is positive correlation? when should we compare DW with DL and Du In this case, since DW=1 falls between DL and DU, so we cannot infer any statistical result about Serial Correlation. > if 1st, 2nd 3rd AR are not significant and 4th lags is statistically significant, what should we do? Then the 4th lagged variable becomes the ONLY independent variable in regression equation.

man, rus1Bus you have the patience of a saint, I wouldve put this guy on ignore from the beginning.