how do I debounce the @Output of an inner component?

To debounce values you could use a Subject. A subject is both an observable and a observer. This means you can treat it as an observable and pass values to it as well.

You could leverage this to pass the new values from the inner-component to it and debounce it this way.

export class DebouncedComponent {
  @Output() outerValueChanged: new EventEmitter<string>();
  const debouncer: Subject<string> = new Subject<string>();

  constructor() {
      // you listen to values here which are debounced
      // on every value, you call the outer component
      debouncer
        .debounceTime(100)
        .subscribe((value) => this.outerValuedChanged.emit(value));
  }

  onInnerChange(value) {
    // send every value from the inner to the subject
    debouncer.next(value);
  }
}

This is untested pseudo-code. You can see a working example of the concept here (http://jsbin.com/bexiqeq/15/edit?js,console). It's without angular but the concept remains the same.

Update: For newer versions of Angular you might need a slight: change debouncer.debounceTime(100) gets changed to debouncer.pipe(debounceTime(100))

constructor() {
      // you listen to values here which are debounced
     // on every value, you call the outer component
     debouncer
       .pipe(debounceTime(100))
       .subscribe((value) => this.outerValuedChanged.emit(value));
}

Here is a working example Class with all needed imports, Angular 4+, TypeScript and tslint-friendly :) thought it might help some people looking for what I was looking moments ago!

import { Component, Input, Output, EventEmitter, ViewChild, OnDestroy } from '@angular/core';
import { Subject } from 'rxjs/Subject';
import { Subscription } from 'rxjs/Subscription';
import 'rxjs/add/operator/debounceTime';

@Component({
  selector: 'app-searchbox',
  template: `
    <input type="text" autocomplete="off" [(ngModel)]="query" [placeholder]="placeholder" (change)="onInputChange($event)">
  `
})
export class SearchboxComponent implements OnDestroy {
  @Input() debounceTime = 500;
  @Output() change = new EventEmitter<string>();

  query = '';
  placeholder = 'Search...';
  results;

  debouncer = new Subject<string>();
  subs = new Array<Subscription>();

  constructor() {
    this.subs.push(this.debouncer.debounceTime(this.debounceTime).subscribe(
      (value: string) => { this.change.emit(value); },
      (error) => { console.log(error); }
    ));
  }

  onInputChange(event: any) {
    this.debouncer.next(event.target.value);
  }

  ngOnDestroy() {
    for (const sub of this.subs) {
      sub.unsubscribe();
    }
  }
}