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Year 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1-(Answered)

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Question

Three Questions of Stat course-Time series?

Need to use R to finish these three questions

Clearer vision for the questions and data are also attached in the drop box



1. Consider the series annual precipitation in "Lake Michigan" (yearly, 1900-
1986).
(a) Plot the series, obtain its ACF and PACF plots. What is your tentative
conclusion about the type of model that may be suitable for this data?
(b) Fit AR(p) models to this data, p = 0; ::; 5. Select the appropriate one by
using the AICC criterion.
(c) For the selected model in part (b), obtain the parameter estimates and their
standard errors. Plot the residuals. Also obtain the plots of ACF and PACF of
the residuals (15 lags). Do these plots suggest that the model selected in part
(b) a reasonable one? Explain.


2. For each of the following models, generate a mean zero random series of
length n = 500 with 2 = 1 (R function arima.sim), plot the data, and obtain
the ACF and PACF plots (with 25 lags). Comment on the results in each case.
(a) MA(1); = 0:8, (b) MA(2), 1 = 0:4; 2 = ????0:6;
(c) AR(2) with 1 = 0:5; 2 = ????0:6, (d) AR(2) with 1 = ????0:3; 2 = 0:6;
(d) ARMA(1; 2) with = 0:6, 1 = 0:4 and 2 = ????0:6:


3. Let fXtg be a stationary sequence (not necessarily AR(p)) with mean and
autocovariance function . Let us denote (1) by . Show that
(a) the best linear predictor of Xt using Xt????1 is X(f)
t = + (Xt????1 ???? ),
(b) the best linear predictor of Xt????2 using Xt????1 is X(b)
t????2 = + (Xt????1 ???? ),
(c) E(Xt ???? X(f)
t )2 = (1 + 2)(0) ???? 2(1);
(d) E(Xt????2 ???? X(b)
t????2)2 = (1 + 2)(0) ???? 2(1);
(e) Cov(Xt ???? X(f)
t ;Xt????2 ???? X(b)
t????2) = (2) ???? 2(1) + 2(0);
(f) the partial correlation between Xt and Xt????2 given Xt????1 is given by
[(2) ???? 2(1) + 2]=[1 + 2 ???? 2(1)]:




Year

 

1900

 

1901

 

1902

 

1903

 

1904

 

1905

 

1906

 

1907

 

1908

 

1909

 

1910

 

1911

 

1912

 

1913

 

1914

 

1915

 

1916

 

1917

 

1918

 

1919

 

1920

 

1921

 

1922

 

1923

 

1924

 

1925

 

1926

 

1927

 

1928

 

1929

 

1930

 

1931

 

1932

 

1933

 

1934

 

1935

 

1936

 

1937

 

1938

 

1939

 

1940

 

1941

 

1942

 

1943

 

1944

 

1945

 

1946

 


 

Annual precipitation (inches), Lake Michigan, 1900 to 1986

 

33.2

 

27.55

 

31.81

 

32.5

 

31.01

 

34.83

 

33.19

 

31.56

 

29.29

 

32.62

 

26.14

 

35.26

 

32.9

 

29.67

 

31.57

 

31.21

 

34.55

 

27.15

 

29.99

 

30.53

 

29.02

 

31.58

 

32.82

 

28.86

 

32.13

 

25.56

 

34

 

32.35

 

32.83

 

30.56

 

22.21

 

30.95

 

28.44

 

29.86

 

28.04

 

30

 

28.39

 

30.1

 

36.21

 

29.47

 

34.45

 

33.95

 

35.62

 

32.27

 

29.5

 

36.61

 

27.86

 


 

1947

 

1948

 

1949

 

1950

 

1951

 

1952

 

1953

 

1954

 

1955

 

1956

 

1957

 

1958

 

1959

 

1960

 

1961

 

1962

 

1963

 

1964

 

1965

 

1966

 

1967

 

1968

 

1969

 

1970

 

1971

 

1972

 

1973

 

1974

 

1975

 

1976

 

1977

 

1978

 

1979

 

1980

 

1981

 

1982

 

1983

 

1984

 

1985

 

1986

 


 

33.34

 

29.1

 

30.57

 

32.96

 

37.02

 

30.11

 

28.99

 

36.89

 

27.55

 

27.2

 

30.9

 

25.99

 

37.82

 

34.63

 

31.43

 

26.97

 

24.92

 

29.56

 

36.89

 

29.45

 

33.33

 

34.29

 

31.56

 

32.86

 

30.1

 

34.97

 

33.98

 

32.77

 

35.74

 

26.54

 

34.42

 

33.67

 

33.18

 

32.11

 

31.51

 

32.86

 

31.55

 

29.74

 

38.98

 

34.65

 


 

Annual precLake Mich 1900 to 1986

 


 

 

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