Please help me rephrase this short paragraph and if possible, include theTheEconometricsofPhillipsCurve.pdf in the data finding, just describe their finding in your own wors, thanks

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King’s finding of the new IS-LM models by adding Phillip’s curve phenomenon on inflation expectation.
The forward-looking IS equation makes current real spending yt depend on the expected future level of real spendingEtyt+1 and the real interest rate rt . There is also an aggregate demand shock xdt : a positive xdt raises aggregate spending at given levels of the endogenous determinants Etyt+1 and rt .5
IS : yt = Etyt+1 – s[rt – r] + xdt ————————————————————equation 4
The parameters > 0 determines the effect of the real interest rate on aggregate demand: If s is larger than a given rise in the real interest rate causes a larger decline in real demand. The parameter r > 0 represents the rate of interest which would prevail in the absence of output growth and aggregate demand shocks. The new IS equation is described as forward-looking because Etyt+1 enters on the right-hand side.
The Fisher equation makes the nominal interest rate Rt equal to the sum of the real interest rate rt and the rate of inflation that is expected to prevail between t and t+1, Etpt+1.
F : Rt = rt + Etpt+1 ————————————————————————-equation 5
This conventional specification of the Fisher equation omits any inflation risk premium in the nominal interest rate.
The expectational Phillips curve relates the current inflation rate pt to expected future inflation Etpt+1, the gap between current output yt and capacity output yt , and an inflation shock xpt
PC : pt = ßEtpt+1 + ?(yt – yt ) + xpt——————————————-equation 6
The parameter ß satisfies 0 = ß = 1. The parameter ? > 0 governs how inflation responds to deviations of output from the capacity level. If there is a larger value of ? then there is a greater effect of output on inflation; in this sense, prices may be described as adjusting faster—being more flexible—if ? is greater.
Using the definition of the inflation rate pt =…

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