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The data file (X in code thread below) contains the record of monthly data X[t] over a twenty year period.

The data can be modelled by X[12j+i] = Mu + s[i] + Y[12j+i] where (i=1,...,12; j=1,...,k) where Mu, s[1],...,s[12] are parameters of the model, Z[t] is white noise WN(0,sigma^2) and k=20. Given the least-squares estimators of Mu and Mu+s[i] are the overall mean and the mean of all observations recorded in the ith period, respectively. Obtain a least-squares fit of this model to the data.

X <- c(20.73,20.51,21.04,21.34,21.60,21.67,21.93,22.18,21.55,21.38,20.78,20.75,20.57,20.09,20.61,21.33,21.72,21.83,21.70,22.62,21.40,21.53,20.71,20.82,20.73,20.65,20.67,21.40,21.21,21.63,21.77,22.20,21.29,21.44,21.01,20.75,20.64,20.24,21.03,21.61,21.46,21.61,22.08,22.66,21.21,20.95,20.88,20.37,20.53,20.30,21.26,21.14,21.99,21.88,22.46,22.31,21.65,21.60,20.62,20.71,20.64,20.94,20.89,21.19,21.57,21.91,21.71,21.95,21.52,21.06,20.77,20.50,20.67,20.77,21.06,21.70,20.73,21.83,21.71,22.99,21.81,20.97,20.72,20.43,20.49,20.33,20.95,21.34,21.61,21.88,22.28,22.33,21.16,21.00,21.07,20.59,20.87,20.59,21.06,21.23,21.59,21.80,21.76,22.48,21.91,20.96,20.83,20.86,20.36,20.48,20.89,21.35,21.80,21.87,22.13,22.54,21.91,21.33,21.18,20.67,20.98,20.77,21.22,21.09,21.37,21.71,22.45,22.25,21.70,21.67,20.59,21.12,20.35,20.86,20.87,21.29,21.96,21.85,21.90,22.10,21.64,21.56,20.46,20.43,20.87,20.38,21.05,20.78,21.99,21.59,22.29,22.23,21.70,21.12,20.69,20.47,20.42,20.51,21.10,21.39,21.98,21.86,22.40,22.04,21.69,21.32,20.74,20.51,20.21,20.29,20.64,21.29,22.03,21.90,22.22,22.07,21.95,21.57,21.01,20.27,20.97,20.64,20.95,21.19,22.02,21.73,22.35,22.45,21.50,21.15,21.04,20.28,20.27,20.48,20.83,21.78,22.11,22.31,21.80,22.52,21.41,21.13,20.61,20.81,20.82,20.42,21.20,21.19,21.39,22.33,21.91,22.36,21.53,21.53,21.12,20.37,21.01,20.23,20.71,21.17,21.63,21.89,22.34,22.23,21.45,21.32,21.05,20.90,20.80,20.69,20.49,21.28,21.68,21.98,21.67,22.63,21.77,21.36,20.61,20.83)

I found the least squares estimator of Mu and (Mu+s[i])

lse.Mu <- mean(X)
IndicatorVar <- rep(1:12,20)
lse.Mu.si <- c(1:12)
for(i in 1:12){lse.Mu.si[i] <- mean(X[IndicatorVar==i])

This is where I get confused. I'm not sure what to do next to find the least squares fit of the model. I tried finding the estimator of Y:

est.Y <- c(1:240)
for(i in 1:12){for(j in 0:19){est.Y[(12*j)+i] <- X[(12*j)+i] - lse.Mu.si[i]}}

But still I don't know how to use it to find least squares fit or where the Z[t] white noise comes into it.

Can you please help point me in the right direction or let me know what code to use? Iv spent three days on google and I still cant work it out!

Following on from this I need to examine the validity of the model by making a graphical comparison of the data with the model and employ any statistical test that are considered appropriate. Any suggestions on which graphs and statistical tests would be best to use would be greatly appreciated.

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