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Properties of plims

These properties are derived from Slutsky's Theorem.

Let plim Xn=θ\text{plim }X_n=\theta and plim Yn=ψ\text{plim }Y_n=\psi. Let cc be a constant. Then,

  1. plim c=c\text{plim }c=c.
  2. plim Xn+Yn=θ+ψ\text{plim }X_n + Y_n = \theta + \psi.
  3. plim XnYn=θψ\text{plim }X_n Y_n = \theta \psi.
  4. plim cXn=cθ\text{plim }cX_n= c\theta.
  5. plim XnYn=θψ. (provided ψ≠0)\text{plim }\frac{X_n}{Y_n} = \frac{\theta}{\psi}.\text{ (provided }\psi\neq0)
  6. plim g(Xn,Yn)=g(θ,ψ). (assuming it exists and g(.) is cont. diff.)\text{plim }g(X_n, Y_n) = g(\theta,\psi).\text{ (assuming it exists and g(.) is cont. diff.})

We can generalize Slutsky’s Theorem to matrices.

Let plim An=A\text{plim }\bold{A}_n=\bold{A} and plim Bn=B\text{plim }\bold{B}_n=\bold{B} (element by element). Then,

  1. plim An−1=[plim An]−1=A−1.\text{plim }\bold{A}_n^{-1}=[\text{plim }\bold{A}_n]^{-1}=\bold{A}^{-1}.
  2. plim (AnBn)=plim (An) plim (Bn)=AB\text{plim }(\bold{A}_n\bold{B}_n)=\text{plim }(\bold{A}_n)\text{ plim }(\bold{B}_n)=\bold{AB}.