Chapter 8: Regression with Lagged Explanatory Variables

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1 Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One variable can influence anoher wih a ime lag. 2. If he daa are nonsaionary, a problem known as spurious regression may arise. You will no undersand 2. a his sage. In his chaper, we focus on 1. Assume daa are saionary (explain laer wha his means). 1

2 The Regression Model wih Lagged Explanaory Variables Y = α + β 0 X + β 1 X β q X -q + e Muliple regression model wih curren and pas values (lags) of X used as explanaory variables. q = lag lengh = lag order OLS esimaion can be carried ou as in Chapers 4-6. Saisical mehods same as in Chapers 4-6. Verbal inerpreaion same as in Chaper 6. Ex. β 2 measures he effec of he explanaory variable 2 periods ago on he dependen variable, ceeris paribus. 2

3 Aside on Lagged Variables X is he value of he variable in period. X -1 is he value of he variable in period -1 or lagged one period or lagged X. Defining X and lagged X in a spreadshee X lagged X X 2 X 1 X 3 X 2 X 4 X X T X T-1 Each column will have T-1 observaions. In general, when creaing X lagged q periods you will have T-q observaions. 3

4 Example: Lagged Variables T = 10 Y + = α + β X + β X + β X + β X e Col. A Col. B Col. C Col. D Col. E Y X X lagged 1 period X lagged 2 periods X lagged 3 periods Row 1 Y 4 X 4 X 3 X 2 X 1 Row 2 Y 5 X 5 X 4 X 3 X 2 Row 3 Y 6 X 6 X 5 X 4 X 3 Row 4 Y 7 X 7 X 6 X 5 X 4 Row 5 Y 8 X 8 X 7 X 6 X 5 Row 6 Y 9 X 9 X 8 X 7 X 6 Row 7 Y 10 X 10 X 9 X 8 X 7 4

5 Example: Long Run Predicion of a Sock Marke Price Index The issue of wheher sock marke reurns are predicable is a very imporan (bu difficul) one in finance. This is no a book on financial heory and, hence, we will no describe he heoreical model which moivaes his example. Variables: sock prices, dividends and reurns. The basic equaion relaing hese hree conceps is: ( P P + D ) R P, 1 Reurn = = 100 where R is he reurn on holding a share from period -1 hrough, P is he price of he sock a he end of period D is he dividend earned beween period -1 and. This relaionship, along wih assumpions abou how hese variables evolve in he fuure, can be used o develop various heoreical financial models. One example: he raio of dividends o sock price should have some predicive power for fuure reurns, paricularly a long horizons. 1 5

6 How does such a heory relae o our regression model wih lagged explanaory variables? Dependen variable (Y) is he oal reurn on he sock marke index over a fuure period bu he explanaory variable (X) is he curren dividend-price raio. Y = α + βx + e + h h is forecas horizon + h Y +h is calculaed using he reurns R +1, R +2,.., R +h. Equivalenly:, Y + + = α βx h e. This is a specialized version of he regression model wih lagged explanaory variables. 6

7 Financial heory suggess ha he explanaory power for his regression should be poor a shor horizons (e.g. h=1 or 2) bu improve a longer horizons. Our daa (monhly) Y = welve monh reurns (i.e. h=12) from a sock marke X = dividend-price raio (welve monhs ago). Coeff Sa P-value Lower 95% Upper 95% Iner X E Dividend-price raio does have significan explanaory power for welve monh reurns (since P- value less han.05). Theory ha dividend-price raio has some predicive power for long run reurns is suppored. However, R 2 =0.019 indicaing ha his predicive power is weak. Only 1.9% of he variaion in welve monh reurns can be explained by he dividend-price raio. 7

8 Example: The Effec of Bad News on Marke Capializaion Moivaion: Share price of a company can be sensiive o bad news. E.g. Company B is in an indusry which is paricularly sensiive o he price of oil. If he price of oil goes up, hen he profis of Company B will end o go down and some invesors, anicipaing his, will sell heir shares in Company B driving is price (and marke capializaion) down. However, his effec migh no happen immediaely so lagged explanaory variables migh be appropriae. 8

9 Monhly daa for 5 years (i.e. 60 monhs) on he following variables: Y = marke capializaion of company ($000) X = price of oil (dollars per barrel) e X X X X X Y = β β β β β α 9

10 Example: The Effec of Bad News on Marke Capializaion (con.) Resuls: Coeff. S. Err. Sa P-val Lower 95% Upper 95% Iner E X X E X E X X

11 Example: The Effec of Bad News on Marke Capializaion (con.) Wha can he company conclude abou he effec of he price of oil on is marke capializaion? Increasing he price of oil by $1 per barrel in a given monh is associaed wih: An immediae reducion in marke capializaion of $145,000, ceeris paribus. A reducion in marke capializaion of $462,140 one monh laer, ceeris paribus. A reducion in marke capializaion of $424,470 wo monhs laer, ceeris paribus. A reducion in marke capializaion of $199,550 hree monhs laer, ceeris paribus. A reducion in marke capializaion of $36,900 four monhs laer, ceeris paribus. 11

12 Example: The Effec of Bad News on Marke Capializaion (con.) Inuiion abou wha he ceeris paribus condiion implies: Increasing he oil price by one dollar in a given monh will end o reduce marke capializaion in he following monh by $462,120, assuming ha no oher change in he oil price occur. Toal effec = $145,000 + $462,140 + $424,470 + $199,550 + $36,900 = $1,268,060 Afer four monhs, he effec of adding one dollar o he price of oil is o decrease marke capializaion by $1,268,

13 Selecion of Lag Order How o choose q (lag lengh)? One way: Use -ess discussed in Chaper 5 sequenially (anoher way is o use informaion crieria which we will discuss laer). Sep 1 Choose he maximum possible lag lengh, qmax, ha seems reasonable o you. Sep 2 Esimae he model: Y + = α + β X + β X β X e qmax qmax If he P-value for esing β qmax =0 is less han he significance level you choose (e.g..05) hen go no furher. Use qmax as lag lengh. Oherwise go on o he nex sep.. 13

14 Sep 3 Selecion of Lag Order (con.) Esimae he model: Y = α + β X + β X β X + e 1 1 qmax 1 qmax If he P-value for esing β qmax-1 =0 is less han he significance level you choose (e.g..05) hen go no furher. Use qmax-1 as lag lengh. Oherwise go on o he nex sep. Sep 4. Esimae he model: Y = α + β X + β X β X + e 1 1 qmax 2 qmax If he P-value for esing β qmax-2 =0 is less han he significance level you choose (e.g..05) hen go no furher. Use qmax-2 as lag lengh. Oherwise go on o he nex sep, ec.. 14

15 Aside: Lag Lengh The number of observaions used in a model wih lagged explanaory variables is equal o he original number of observaions, T, minus he maximum lag lengh. In Sep 2 you have T-qmax observaions In Sep 3, T-qmax+1 observaions, ec. 15

16 Example: The Effec of Bad News on Marke Capializaion (con.) Suppose qmax=4 P-value for X -4 =.44>.05 (see previous able) Drop X -4 and re-esimae wih q = 3. Coeff. S. Err. Sa P-value Lower 95% Upper 95% Iner E X X E X E X P-value for X -3 is.0003 <.05. Selec q=3 and presen his able in a repor. 16

17 Chaper Summary 1. Regressions wih ime series variables involve wo issues we have no deal wih in he pas. Firs, one variable can influence anoher wih a ime lag. Second, if he variables are nonsaionary, he spurious regressions problem can resul. The laer issue will be deal wih laer on. 2. Disribued lag models have he dependen variable depending on an explanaory variable and lags of he explanaory variable. 3. If he variables in he disribued lag model are saionary, hen OLS esimaes are reliable and he saisical echniques of muliple regression (e.g. looking a P-values or confidence inervals) can be used in a sraighforward manner. 4. The lag lengh in a disribued lag model can be seleced by sequenially using -ess beginning wih a reasonably large lag lengh. 17

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