Assignment No 10

Assignment 10

Problem 1:-

Create 3 vectors, x, y, z and choose any random values for them, ensuring they are of equal length,
T<- cbind(x,y,z)
Create 3 dimensional plot of the same (all 3 types)

Solution:-

Commands:-

> sample<-rnorm(50,25,6)

> x<-sample(sample,10)

> y<-sample(sample,10)

> z<-sample(sample,10)

> T<-cbind(x,y,z)

 Screenshots:-

> plot3d(T)

plot3d(T,col=rainbow(1000))

plot3d(T,col=rainbow(1000),type=’s')

Problem 2:-

Read the documentation of rnorm and pnorm and 

Create 2 random variables
Create 3 plots:
1. X-Y
2. X-Y|Z (introducing a variable z and cbind it to z and y with 5 diff categories) Hint: ?factor
3. Color code and draw the graph
4. Smooth and best fit line for the curve

Solution:-

Commands:-

> x<-rnorm(200,mean=5,sd=1)
> y<-rnorm(200,mean=3,sd=1)
> z1<-sample(letters,5)
> z2<-sample(z1,200,replace=TRUE)
> z<-as.factor(z2)
> t<-cbind(x,y,z)

Screenshots:-

> qplot(x,y)

> qplot(x,z,alpha=I(2/10))

> qplot(x,z)

> qplot(x,y,geom=c(“point”,”smooth”))

> qplot(x,y,colour=z)

> qplot(log(x),log(y),colour=z)

Assignment No 9

ITBAL Assignment 9

Analysis of:

DIPITY

Assignment 8

Assignment - Panel Data Analysis

We will be doing Panel Data Analysis of "Produc" data

We will be analysing on three types of model :
      Pooled affect model
      Fixed affect model
      Random affect model

Then we will be determining which model is the best by using functions:
       pFtest : for determining between fixed and pooled
       plmtest : for determining between pooled and random
       phtest: for determining between random and fixed

> data(Produc , package ="plm")

> head(Produc)

    state year     pcap     hwy   water    util       pc   gsp    emp unemp

1 ALABAMA 1970 15032.67 7325.80 1655.68 6051.20 35793.80 28418 1010.5   4.7

2 ALABAMA 1971 15501.94 7525.94 1721.02 6254.98 37299.91 29375 1021.9   5.2

3 ALABAMA 1972 15972.41 7765.42 1764.75 6442.23 38670.30 31303 1072.3   4.7

4 ALABAMA 1973 16406.26 7907.66 1742.41 6756.19 40084.01 33430 1135.5   3.9

5 ALABAMA 1974 16762.67 8025.52 1734.85 7002.29 42057.31 33749 1169.8   5.5

6 ALABAMA 1975 17316.26 8158.23 1752.27 7405.76 43971.71 33604 1155.4   7.7

> pool <- plm(log(pcap)~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) + log(emp) + log(unemp) , data =Produc, model=("pooling"), index = c("state","year"))

> summary(pool)

Oneway (individual) effect Pooling Model

Call:

plm(formula = log(pcap) ~ log(hwy) + log(water) + log(util) +

    log(pc) + log(gsp) + log(emp) + log(unemp), data = Produc,

    model = ("pooling"), index = c("state", "year"))

Balanced Panel: n=48, T=17, N=816

Residuals :

    Min.  1st Qu.   Median  3rd Qu.     Max.

-0.04950 -0.01940 -0.00412  0.01150  0.08690

Coefficients :

              Estimate Std. Error  t-value  Pr(>|t|)   

(Intercept)  0.7496721  0.0271054  27.6577 < 2.2e-16 ***

log(hwy)     0.5248704  0.0048326 108.6099 < 2.2e-16 ***

log(water)   0.1077579  0.0040454  26.6370 < 2.2e-16 ***

log(util)    0.4127255  0.0038337 107.6574 < 2.2e-16 ***

log(pc)     -0.0330829  0.0048219  -6.8610 1.361e-11 ***

log(gsp)     0.0758341  0.0108650   6.9797 6.170e-12 ***

log(emp)    -0.0891772  0.0076891 -11.5978 < 2.2e-16 ***

log(unemp)   0.0043878  0.0029465   1.4891    0.1368   

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:    724.14

Residual Sum of Squares: 0.56734

R-Squared      :  0.99922

      Adj. R-Squared :  0.98942

F-statistic: 147217 on 7 and 808 DF, p-value: < 2.22e-16

Fixed Affect Model:

> fixed <- plm(log(pcap)~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) + log(emp) + log(unemp) , data =Produc, model=("within"), index = c("state","year"))

> summary(fixed)

Oneway (individual) effect Within Model

Call:

plm(formula = log(pcap) ~ log(hwy) + log(water) + log(util) +

    log(pc) + log(gsp) + log(emp) + log(unemp), data = Produc,

    model = ("within"), index = c("state", "year"))

Balanced Panel: n=48, T=17, N=816

Residuals :

     Min.   1st Qu.    Median   3rd Qu.      Max.

-0.069800 -0.005280 -0.000327  0.005360  0.061200

Coefficients :

             Estimate Std. Error t-value  Pr(>|t|)   

log(hwy)    0.5418395  0.0109565 49.4536 < 2.2e-16 ***

log(water)  0.1215676  0.0053719 22.6304 < 2.2e-16 ***

log(util)   0.3909247  0.0065771 59.4368 < 2.2e-16 ***

log(pc)     0.0177190  0.0096372  1.8386 0.0663624 . 

log(gsp)    0.0568433  0.0126569  4.4911 8.184e-06 ***

log(emp)   -0.0851515  0.0146508 -5.8121 9.073e-09 ***

log(unemp) -0.0092135  0.0024988 -3.6872 0.0002429 ***

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:    9.4468

Residual Sum of Squares: 0.12613

R-Squared      :  0.98665

      Adj. R-Squared :  0.92015

F-statistic: 8033.41 on 7 and 761 DF, p-value: < 2.22e-16

Random Affect Model:

> random <- plm(log(pcap)~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) + log(emp) + log(unemp) , data =Produc, model=("random"), index = c("state","year"))

> summary(random)

Oneway (individual) effect Random Effect Model

   (Swamy-Arora's transformation)

Call:

plm(formula = log(pcap) ~ log(hwy) + log(water) + log(util) +

    log(pc) + log(gsp) + log(emp) + log(unemp), data = Produc,

    model = ("random"), index = c("state", "year"))

Balanced Panel: n=48, T=17, N=816

Effects:

                    var   std.dev share

idiosyncratic 0.0001657 0.0128743 0.221

individual    0.0005848 0.0241825 0.779

theta:  0.8719 

Residuals :

    Min.  1st Qu.   Median  3rd Qu.     Max.

-0.06500 -0.00624 -0.00195  0.00454  0.06450

Coefficients :

              Estimate Std. Error t-value  Pr(>|t|)   

(Intercept)  0.6625006  0.0530786 12.4815 < 2.2e-16 ***

log(hwy)     0.5021294  0.0074551 67.3537 < 2.2e-16 ***

log(water)   0.1191683  0.0049801 23.9289 < 2.2e-16 ***

log(util)    0.3944635  0.0060802 64.8768 < 2.2e-16 ***

log(pc)      0.0101901  0.0075870  1.3431    0.1796   

log(gsp)     0.0599363  0.0122997  4.8730 1.323e-06 ***

log(emp)    -0.0767378  0.0125556 -6.1119 1.531e-09 ***

log(unemp)  -0.0034020  0.0022591 -1.5059    0.1325   

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:    21.167

Residual Sum of Squares: 0.13965

R-Squared      :  0.9934

      Adj. R-Squared :  0.98366

F-statistic: 17380.4 on 7 and 808 DF, p-value: < 2.22e-16

Comparison:

The comparison between the models would be a Hypothesis testing based on the 

following concept: 

H0: Null Hypothesis: the individual index and time based params are all zero

H1: Alternate Hypothesis: atleast one of the index and time based params is non zero 

Pooled vs Fixed

Null Hypothesis: Pooled Affect Model

Alternate Hypothesis : Fixed Affect Model

> pFtest(fixed,pool)

       F test for individual effects

data:  log(pcap) ~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) +      log(emp) + log(unemp)

F = 56.6361, df1 = 47, df2 = 761, p-value < 2.2e-16

alternative hypothesis: significant effects

Since the p value is negligible so we reject the Null Hypothesis and hence Alternate hypothesis is accepted which is to accept Fixed Affect Model.

Pooled vs Random

Null Hypothesis: Pooled Affect Model

Alternate Hypothesis: Random Affect Model

> plmtest(pool)

        Lagrange Multiplier Test - (Honda)

data:  log(pcap) ~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) +      log(emp) + log(unemp)

normal = 57.1686, p-value < 2.2e-16

alternative hypothesis: significant effects

Since the p value is negligible so we reject the Null Hypothesis and hence Alternate hypothesis is accepted which is to accept Random Affect Model.

Random vs Fixed

Null Hypothesis: No Correlation . Random Affect Model

Alternate Hypothesis: Fixed Affect Model

> phtest(fixed,random)

        Hausman Test

data:  log(pcap) ~ log(hwy) + log(water) + log(util) + log(pc) + log(gsp) +      log(emp) + log(unemp)

chisq = 93.546, df = 7, p-value < 2.2e-16

alternative hypothesis: one model is inconsistent

Since the p value is negligible so we reject the Null Hypothesis and hence Alternate hypothesis is accepted which is to accept Fixed Affect Model.

Conclusion: 

So after making all the comparisons we come to the conclusion that Fixed Affect Model is best suited to do the panel data analysis for "Produc" data set.

Hence , we conclude that within the same id i.e. within same "state" there is no variation.

Assignment No 6

Stationary Time Series

Q. Use NSE Nifty Index (Feb 6, 2012 to Feb 6, 2013)
a. Create log of the return data.

b. Calculate Historical volatility.

c. Analyse Stationarity by-

    1. Create Auto Correlation Function Plot.

    2. Augmented Dickey Fuller Test.

Solution:

> nse1yr<-read.csv(file.choose(),header=T)

> closingprice<-nse1yr$Close

> closingprice.ts<-ts(closingprice,frequency=252)

> laggingtable<-cbind(closingprice.ts,lag(closingprice.ts,k=-1)

+ ,closingprice.ts-lag(closingprice.ts,k=-1))

> Return<-(closingprice.ts-lag(closingprice.ts,k=-1))/lag(closingprice.ts,k=-1)

> head(Return)

[1] -0.004942508  0.006185393  0.008233749 -0.005681451  0.001598038

[6]  0.004795740

> Return.Std<-scale(Return)+10

> logreturn<-log(Return.Std)

> acf(logreturn)

From the figure it implies that the all the standard errors are within the 95% confidence interval and hence we can say that the time series is stationary.

> T<-252^.5

> Historicalvolatility<-sd(Return)*T

> Historicalvolatility

[1] 0.1424954

> adf.test(logreturn)

        Augmented Dickey-Fuller Test

data:  logreturn

Dickey-Fuller = -6.2635, Lag order = 6, p-value = 0.01

alternative hypothesis: stationary

Warning message:

In adf.test(logreturn) : p-value smaller than printed p-value

 

Since p-value is less than 0.05 ,therefore we can say null hypothesis is rejected and unit root is not 1 hence the time series is stationary so data analysis can be done.