Supplement

Power Computations for Intervention Analysis

The result of Tiao et al. (1990) is shown to be a special case of our results in Table 1 for a ramp intervention with AR(1) noise.

Tiao et a. (1990, Appendix A),  p.20,515

Let (Subscript[ℛ, t]^((T))) denote the ramp intervention defined in Tiao (1990) so we have (Subscript[ℛ, t]^((T))) = 
        (Subscript[R, t]^((T + 1)))/12.   Then assuming T = 0 the following expression was derived for \(TraditionalForm\`σ\_ω\&^\),

\(TraditionalForm\`\(\(σ\_\(ω\&^\)\%\((Tiao)\) = σ\ \(g(T, ϕ)\)\)\(,\)\)\)

where σ = (Subscript[σ, a]^2) /(1 - (ϕ^2)) and  \(TraditionalForm\`g(T, ϕ) = 
        24\ \((3\ \((1 + ϕ)\)/\((\((1 - ϕ)\) T \((T\^2 - 1)\))\))\)\^\(1/2\)\)

In Mathematica this expression can be defined as:

24 (((3 (1 + ϕ)/((1 - ϕ) T ((T^2) - 1))))^(1/2)) // Simplify

gTiao = %

Comparison With Our Result

Let \(TraditionalForm\`\(σ\_\(ω\&^\)\)(T, ϕ)\) denote our estimate.  Then taking into account (Subscript[ℛ, t]^((T))) = 
        (Subscript[R, t]^((T + 1)))/12.   We obtain, \(TraditionalForm\`𝔤(
          T, ϕ) = \@\(\((1 - ϕ\^2)\)/12\)\ \ \(\(σ\_\(ω\&^\)\)(T + 1, ϕ)\)\)as the corresponding value in our notation.  In order to compare with our result we must assume that the mean is known.

Definition of Mathematica Function

The results in our Table 1 may be written as:

InformationMatrixAR1Ramp[n_, 
        T_,  ϕ_] := 
With[{κ = ϕ - 
              1, 
b12 = (1 - ϕ) (n + 1 - 
                  T) (n + 2 - T - ϕ (n - T))/
                2, 
b22 = 
            (1/6) (1 + n - T) (6 + 7 n + 2 (n^2) - 7 T - 4 n T + 
                  2 (T^2) - 8 n ϕ - 4 (n^2) ϕ + 8 T ϕ + 
                  8 n T ϕ - 4 (T^2) ϕ + n (ϕ^2) + 
                  2 (n^2) (ϕ^2) - T (ϕ^2) - 4 n T (ϕ^2) + 
                  2 (T^2) (ϕ^2))}, {{n (κ^2), b12}, {b12, 
            b22}}
]

Special Case

The special case corresponds to mean known so,

g[T_, ϕ_] := (Sqrt[Inverse[
                  InformationMatrixAR1Ramp[T, 1, ϕ]] // Flatten // 
              Last])*Sqrt[1 - (ϕ^2)]*12

g[T, ϕ] // FullSimplify

Proof of Equivalence

Both expressions are positive, so equality of the squared result establishes the equivalence.

gTiao^2 - g[T, ϕ]^2 // FullSimplify